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.
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.
Best Answer
Not quite like in the photo above, which shows more than what the naked eye can see, but yes, absolutely! Our galaxy (well, the chunk of it visible from these parts) is a naked-eye object. The fact that your question even exists shows how much time is now spent by people under light-polluted skies.
It will not be visible from the city, however. You need to drive an hour (or two, if you live in a huge urban area) to the country side, far from city lights. Stay outside in full darkness for a few minutes, then look up. There will be a faint "river" of light crossing the sky. That's the Milky way. Full dark adaptation occurs after 30 minutes of not seeing any source of light, but this is not required for seeing our galaxy.
While you're in a dark sky area, also look up the Andromeda galaxy, a.k.a. M31.
http://www.physics.ucla.edu/~huffman/m31.html
I mean, if you can see M31 with the naked eye, at 2 mil light-years away, then of course you can see Milky Way, which is basically in our backyard.
Here's a light pollution map, not very recent, but still useful:
http://www.jshine.net/astronomy/dark_sky/