There are numerous distance indicators used for within the galaxy. The most common way is by using intrinsic magnitude. By knowing how bright an object would be if we were close, we can determine how far away it is by how dim it is. There are many types of stars where we have a rough idea of how bright they should be due to characteristics of the star:
Cephied Variables: The original type of variable star that was used by Hubble to determine the distance to the Andromeda Galaxy.
RR Lyrae Variable: Like the Cephied variable, but usually dimmer.
Type 1a Supernova: These guys, unlike the first two, are cataclismic variables. Essentially a binary white dwarf slowly accretes matter from its binary till it reaches the Chandrashankar Limit, after which point it explodes in a very characteristic way (since the mass at the time of explosion is roughly constant).
Main Sequence Stars: Generally less accurate than the first 3, there are some types of main sequence stars which are used to find distances in a similar way.
There are a few other ways we can measure distances:
Perpendicular Movement: For example there is a "light echo" from SN 1987A which is essentially light from the supernova interacting with dust around the old star. Since this echo should be expanding at the speed of light, we can tell how far away the nova is by the angular velocity of the light.
Relative Velocity in a Moving Cluster: (see dmckee's answer)
Tulley-Fisher relation: A relationship between the luminosity of the galaxy and it's apparent width. Can be used as a decent distance calculator.
Faber-Jackson Relation: Similar to Tulley-Fisher, relates luminosity with radial velocity dispersion rate.
EDIT: Some more information about redshifts.
The whole relationship between redshift and distance was in fact established by Hubble by relating distance to Cephied variables (I believe) with redshift. Later on it was made more precise using supernova, which are brighter and can be seen from much father away (I think recent supernova can be occasionally seen around Z=2, while Cephieds are all Z<1). Within a galaxy, redshift cannot be used directs since the "peculiar velocity," the velocity within the galaxy, completely overshadows the effects of universe expansion on which Hubble's Law is based. Redshift within the galaxy is useful for certain other techniques.
EDIT: corrected a few minor errors.
The estimates I've read are similar to yours: 200 to 400 billion stars. Counting the stars in the galaxy is inherently difficult because, well, we can't see all of them.
We don't really count the stars, though. That would take ages: instead we measure the orbit of the stars we can see. By doing this, we find the angular velocity of the stars and can determine the mass of the Milky Way.
But the mass isn't all stars. It's also dust, gas, planets, Volvos, and most overwhelmingly: dark matter. By observing the angular momentum and density of stars in other galaxies, we can estimate just how much of our own galaxy's mass is dark matter. That number is close to 90%. So we subtract that away from the mass, and the rest is stars (other objects are more-or-less insignificant at this level).
The mass alone doesn't give us a count though. We have to know about how much each star weighs, and that varies a lot. So we have to class different types of stars, and figure out how many of each are around us. We can extrapolate that number and turn the mass into the number of stars.
Obviously, there's a lot of error in this method: it's hard to measure the orbit of stars around the galactic center because they move really, really slowly. So we don't know exactly how much the Milky Way weighs, and figuring out how much of that is dark matter is even worse. We can't even see dark matter, and we don't really understand it either. Extrapolating the concentrations of different classes of stars is inexact, and at best we can look at other galaxies to confirm that the far side of the Milky Way is probably the same as this one. Multiply all those inaccuracies together and you get a range on the order of 200 billion.
Best Answer
As has been said, this is probably a very subjective question/answer. Not only that, but the composition of galaxies, and even regions within a galaxy, varies a great deal. Then there is the question of what constitutes as being part of the galaxy as opposed to perhaps a small orbiting dwarf galaxy. The answer you got from the Quora seems to be pretty comprehensive.
The volume of an area of interest, divided by the number of stars in that area seems to be the one that most people take as the approach. Which may not get a very accurate result, but smoothed out over said volume. Although, I will note that the first technique given on the quora site gives an answer that is close to the accepted "average" in the Milky Way, so at least there doesn't seem to be a large disagreement there. Of course, that assumes that the same initial starting conditions are used in both problems, which is highly unlikely since they aren't totally agreed upon anyway.
EDIT TO ADD: For more examples of similar math, here Dr. Plait calculates the number of habitable planets (where he shows the calculation for the volume of the galaxy). Making some assumptions of our own (like 200,000,000,000 stars which is LOW in my opinion), we come out to an average distance of about 5 light years. Doubling the number of stars gives an average of about 4 light years though, so again, we are not off by factors.