Celsius was defined by fixing the mercury's expansion coefficient with respect to temperature as constant. It is now defined as Kelvin plus 273.15, not the other way around. In fact, the freezing point of water under standard atmospheric temperature is 0.000089(10)°C, boiling point 99.9839°C. Mercury thermometer is now an approximate measuring device, as any measuring device, rather than the definition of Celsius scale.
The original definition of Celsius and Fahrenheit are arbitrary and artificial, but Kelvin, or thermodynamic temperature, is based on universal physical principle, i.e., second law of thermodynamics. This relation with fundamental physical principle makes Kelvin the "clean" scale of temperature.
Kelvin still depends on water, with its triple point fixed at 0.01°C (273.16K). The proposed redefinition of Kelvin will fix Boltzmann constant, the constant relating temperature and energy. That will make Kelvin even more natural.
The NIST style guide is pretty good — that's a place where people really care about getting details right.
I use lower-case names for spelled-out units, even when named for famous people or having uppercase abbreviations (N -> newton, J -> joule, L -> liter (unless you count $\ell$ for liter), K -> kelvin). I think that "degrees kelvin" is entirely by analogy with "degrees Celsius" and "degrees Fahrenheit" (the latter two of which I think I have always seen capitalized).
I don't pluralize "kelvin" when talking about temperatures: I talk about temperatures like "two hundred fifty millikelvin," or "four kelvin" for the boiling point of helium, or "three hundred kelvin" for room temperature. This is not consistent with the way that I would discuss a length unit, or a mass unit.
I think that this may be because temperature is an intensive variable. If I have a thing that weighs a kilogram, and another thing that weighs four kilograms, and I put them together, I know that I have five kilograms worth of stuff. But if I have some stuff at one kelvin, and some other stuff at four kelvins, and I mix them together, I don't get some stuff at five kelvins. I know this, and so I don't think of "a kelvin" as a lump of temperature that I can carry around and add or subtract to things.
I feel the same way about the hertz: I have no desire to say "sixty hertzes." Combining an oscillator at 60 Hz and an oscillator at 10 Hz gives me something much more complicated than an oscillator at 70 Hz. I notice that "hertz" is listed as one of your three exceptions, though.
If that's really my thought process I would make the same decision about the pascal (for pressure) and the poise (for viscosity); I can't think off the top of my head of another intensive quantity with a named unit. I think that if you asked me I would tell you that air pressure at sea level is "ten to the five pascal," but I'm focusing too hard on it and I'm honestly not sure.
In response to a comment: I definitely
do say things like "two atmospheres of pressure," but never "two bars" or "one thousand torrs." It could well be that dealing with kelvins one by one is so rare that I don't think of them as being countable. Interesting.
Best Answer
In a sense, there is only one fixed point. Look to the definition (http://www.bipm.org/en/CGPM/db/13/4):
There's a lot hiding in that simple definition. One key concept is that of thermodynamic temperature. A temperature scale that does not have zero at absolute zero is not a thermodynamic temperature scale. So in this sense, the Kelvin scale has two fixed points, absolute zero and the triple point of water.
Another issue hiding in that definition is that it uses the triple point of water. Purified water from an East African lake, from the oceans, or Antarctic ice have slightly different triple points, slightly different freezing points, and slightly different boiling points. The isotopic composition of water varies with latitude. To get around this issue, the water used in determining the triple point of water must have a very specific isotopic composition specified by the Vienna Standard Mean Ocean Water (VSMOW) standard.
This portion of your text is extremely outdated. The triple point of water has been used as the sole fixed point in the SI temperature scale since 1954. Using the freezing and boiling point of water as defining temperature is a discarded concept. Textbooks are oftentimes a decade or so out-of-date; that happens all the time. On the other hand, a textbook that is over six decades out-of-date is unacceptable.
Even using the triple point of water as a fixed point will most likely soon be discarded, at least in theory. If all goes according to plan, the only fixed point will be absolute zero. Temperature in the International System of Units will soon be defined by making the Boltzmann constant a defined value.
In practice, measuring temperature is rather difficult. The International Temperature Scale, which attempts to be a practical realization of the SI concept of the kelvin, has 14 different fixed points (excluding absolute zero). The melting and boiling points of water are not amongst those fixed points. The triple point of water (VSMOW) is.