I think I see the heart of your question. It has nothing to do with relativity in fact. Let me attempt to rephrase.
In the past (say 1700’s) we had pendulum clocks to keep time. We said that every tick of the clock was one second. However, sailors at the time realized that if you put a pendulum clock on a boat it would run "fast" or "slow" because the rocking of the boat or temperature variations would alter the physics of the pendulum. How could they tell it was running fast or slow? They could bring the clock back to Greenwich where it was originally set and notice that their clock ticked 100,000 times whereas the Greenwich clock ticked 120,000 times. This was easily explained by what happened to the clock belonging to the sailors on the boat.
Now, your concern is that when people talk about atomic clocks (the modern day "standard" for time-keeping) they do not mention deleterious effects such as "rocking of the boat" that may cause the atomic clocks to run fast or slow. Your concern is that the atomic clock might be undergoing "rocking" but we just sweep it under the rug and just say that "time" is running fast or slow. The question is why the shift in attitude? Previously we recognized physical mechanisms that could alter how the clock runs and admitted they make the clock worse, but now we just cover it up by saying time is running fast or slow. What gives?
I hope the above was an accurate restatement of your question. Let me now provide my answer.
1) First, the title of this post is "What do clocks measure?". You have suggested that clocks measure rates. I think this is incorrect. I believe that clocks measure a NUMBER of events. A pendulum clock measures how many times the pendulum reaches its right extremity. A quartz oscillator measures how many times its tines reache the extremities of their motion. An atomic clock measures how many times the electron wavefunction revolves around the nucleus*. What are rates then? Well, we have defined the second to be something like: Whenever the Cesium clock in Boulder ticks 9,192,631,770 times we say one second of time has passed. Thus, we can now say (based on the definition) that the Cesium clock ticks at a rate of 9,192,631,770 ticks per second. The fundamental measured quantity is a number of events, the defined quantity is a time, and the derived quantity is a rate.
2) Ok. But, just like on the boat, can't deleterious effects affect how quickly the Cesium clock ticks? That is, how much "time" it takes between two ticks might change if the Cesium clock is "rocking". How come I don't hear about that sort of thing? Well, you probably just haven't heard about that sort of thing because you aren't immersed in the field of precision measurement or atomic physics. Atomic physicists in fact worry about things that mess up the ticking rate of their atoms all of the time. Things that can mess up the ticking rate are electric/magnetic fields (cause atoms to tick faster or slower), collisions with other atoms etc. One problem with atomic clocks is that blackbody radiation emitted by the room temperature vacuum chamber in which the atoms reside causes the atoms to change their ticking frequency. Because of all of this it is recognized that the second is defined to be the amount of time it would take a Cesium atom to tick 9,192,631,770 times if it was at 0 K, in 0 magnetic field, in 0 electric field with no external influences. However, physicists realize that this is impossible to achieve in the lab. Nonetheless, there are technological benefits to attempting to do the best they can. So they perform a certain experiment and measure Cesium ticks in a particular way the best they can and report to the world whenever their cesium ticks. Physicists are continually trying to build better clocks that have lower uncertainties so that they can study ever more precise physics and explore new technologies.
3) If clocks can always have some error than what is the benefit to having clocks at all? Well, even though clocks are always wrong to some degree they are also right to some degree. For example, my friend can say to me: "Hey let's meet at the bowling alley after the quartz oscillator in MY watch ticks 34,875,329 times!" and even if he goes to his house (which he keeps at 65 F) and I go to my house (which I keep at 70 F) and drop MY watch in the sink (it is waterproof) I can still pull it out and have faith that once my watch ticks 34,875,329 times my friend’s watch will ALSO have ticked the same amount so I can get to the bowling alley and not annoy him by being 5,328 ticks late!**
The point of making better and better clocks is for humans to be able to have such faith in each other's time keeping devices on ever finer and finer time scales. For example, if the physicists at the atomic clock in Boulder, Colorado do a good job keeping their clock running (with minimal magnetic fields etc.) and the physicists at the atomic clock Paris France do a good job at their clock then the two parties can have faith that even after the passage of a long amount of time***** they will still have counted the same number of ticks of their clocks. This has practical implications when those clocks are used to synchronize different clocks all around the world including those used for satellite GPS and running the global stock markets, both of which rely on measuring very very small differences in time.
4) And another note reminding us of the sailors. Let's ask again how the sailors knew their clock was running fast or slow (other than seeing their crewmates kick the pendulum a few times). They would notice that the sun would not rise when they expected it to according to their clock or they would bring their clock to Greenwich and compare it there. In both cases they are comparing their clock to another oscillatory physical phenomenon. The key is that they are comparing their clock to a phenomenon which is more "stable"*** than the one on their ship. However, these two clocks are also of course susceptible to clock fluctuations. If the temperature changes in Greenwich that would affect their pendulum clock as well, just not as much as the smaller pendulum on the boat beset by the harsh maritime environment.
In addition to stability it is important that a clock standard can be recreated elsewhere and give the same results. That is illustrated by the presence of similar Cs atomic clocks around the world. The beauty is that, if you can control the environment well enough, a Cs atom in Boulder has the same ticking frequency as a Cs atom in Paris. If everyone can then synchronize to these and other clocks in the international system of atomic clocks then we can be confident that we can all agree on the time to a part in $10^{-16}$ or so and this can be useful technologically. As you have identified however, there IS a limit to how precise we can be. This is known and recognized and people are always working to improve this limit.
edit: One more note here. Since clocks measure number of events happening, and time is derived from that measurement, in some sense if the clock at Greenwich slows down or the atomic clock at Boulder slows down it is correct to say that time itself is slowing down because that it how time is defined. However, you are correct to point out that we must recognize this is happening because of undesirable physical effects in our apparatus. That is why we recognize that these clocks only have a finite level of precision and we recognize some level of uncertainty in definition/measurement of time. Building a better clock means pushing down this uncertainty.
5) There is a dimension of your question which does involve special relativity but I think that is in fact the less interesting point. In some sense we can say that relativistic effects are just another external effect that causes the clock to tick differently than the clock in Boulder. What if the clock in Boulder is experiencing special relativistic effects? Well, we can still compare it to the clock in Paris and get results good to some precision. Eventually, to build a better clock, perhaps such effects will need to be controlled. Some effects which limit atomic clocks (cause them to tick differently) now are: Atomic collisions, Blackbody radiation, the lasers used to measure the atoms, stray magnetic fields etc. Someone closer to the atomic clock field could do a better job than me at producing this list.
6) I highly recommend reading the popular science/history book "Longitude" by Dava Sobel about the need for and invention of precision chronometers for naval navigation in the 18th century to get a handle on practical reasons WHY we want a precise clock and what we mean by a precise clock. Perhaps after understanding some concrete real-world situations you will have a better view on some of your abstract questions.
edit2:
7) Important note on clock stability. When I say a clock is stable what do I mean? Well say I have two wristwatches built which were manufactured one after another on the assembly line. If I synchronize them today I can then watch them for a year and see how far off they get. If they get off by 30 seconds in a year then I can calculate a fractional discrepancy.
$$
\frac{30 \text{ s}}{1 \text{ yr}}\approx 10^{−6}
$$
I don't know which clock is more correct (accurate), but I know that they agree to a part per million. That is they have a relative stability of $10^{-6}$. Now the standard Cs atomic clocks are good to a part in $10^{16}$ or so if they are compared against eachother**** Again, we do not know which clock is more correct, but we can say that the atomic clocks are more stable than the wristwatches because they can agree with each other for a longer amount of time
*This is a bit of atomic physics here, but the atoms used in atomic clocks can be thought of as being exactly like pendulums. It is a system which is physically oscillating in space. This can be a topic for another question.
**Though as we all know, having an accurate watch is not a guarantee that one won't be late! Such a guarantee requires, in addition, a certain amount of personal responsibility!
***Where stability can be defined in a technical sense by comparing the ticking rate of one clock to a clock which is more physically controlled or by comparing the ticking rate of two clocks if there is no "better" clock around. See section 7)
****The most stable atomic clocks reported in fact use Sr and are accurate to a part in $10^{18}$ or so but these are not used as the official time standard. Perhaps in the future they will be.
*****Note that these clocks tick almost $10^{10}$ (ten trillion) times per second. At a stability of $10^{16}$ these clocks can run for over 10 days and not get off by one tick.
Best Answer
The issue with measuring at the ends is that the pendulum “dwells” at the end point as it turns around, so that there is a greater spread of time for which it looks “at the extreme point” than for which it looks “at the midpoint”.
This spread of time introduces error, whether you are triggering a stopwatch by hand or having some kind of sensor make the decision.
If you tried recording all of the points and determining the period, you’d get the same result — either you take the time between zero crossings of the position, or you take the time between peaks. To get the time between peaks, you need to take a derivative of the position measurement and find its zero crossings, and taking the derivative of a set of data points introduces error.