Essentially, yes. That's where you get the G (perfect 5th from C, and the strongest overtone aside from another C).
But let me back up a bit. The overtone series is as follows (starting from the lowest note, which is the note you play and hear - the fundamental, and proceeding up an imaginary keyboard. The strongest overtones tend to be the first ones in the series.):
C (fundamental)
C (octave above fundamental)
G (fifth above previous C... etc.)
C
E
G
Bb
C
D
E
It goes on, but the theory goes that we can't hear all that. Actually, people usually can't or don't perceive as far as this a lot of times, I think. Again, hard to say, as sometimes what we "hear" we can't quite define other than in terms of its overall effect on us.
So, yes, in simple terms, we use the harmonic series to determine the notes and what goes where, but it's more than just following nature - it's how the notes interact with each other because of that natural set of vibrations. When you play a fifth, for example, both notes will have overtones going within the tone. Just like your piano that vibrates sympathetically with the overtones (when they're close to the natural ones), the two notes of a fifth will likewise gel with each other where the overtones are close (where they aren't, they are too far away for us to really hear the interaction of the vibrations).
If you tune a perfect fifth (C-G), or a G that corresponds exactly with the G overtone of C, the vibrations won't "fight" with each other - you get a pure, smooth, laser-like sound with no interference. If, however, you tune a G that is vibrating at a slightly different rate than the harmonic (perfect fifth), the vibrations don't mesh - they "fight" with one another and produce "beats" - a sort of "wah-wah-wah" sound at the level at which the tones are "fighting" - in this case, at the G.
Not everyone hears beats, specifically, but we do perceive something - sometimes this sound is described as rough, or wobbly, or muddy. The further away from matching the G is from the harmonic, the faster the beats go, until you lose track of the beats and it just sounds, well, melodically out of tune.
So, lets take the third, now, which is a little more complex. You have to go up the series a bit further to hear where the beats are happening, but they happen. I'll try to illustrate below:
C
E
C
E
G
B
C
E--E
This last E is where you will, theoretically, hear the note beating if it is close to, but not quite, a pure third. Again, you may perceive this beating somewhere else or as a different sort of sound or sensation, but this is where it's happening. This is where I learn to listen, as a tuner, at least at first. After a while, you can kind of back up again and just hear the purity of the interval without having to seek out the beats at a particular overtone.
So why not tune everything pure? Because it doesn't work out evenly. Nature is "flawed" and if you tune a circle of perfect fifths, you'd expect to come back around to your original tone, but you don't. Therein lies the conundrum that creates the need for tempering, or adjusting the notes so that this "leftover" interval is absorbed in a strategic way in the span of an octave to make music (or, at least keyboard music) possible. Early temperaments didn't temper the notes much in the base keys, and they tended to stick to these keys and enjoy the beauty of it. Equal temperament makes all keys sound exactly the same by making a mathematically "perfect" division of tone, but in the process (as I said originally) it makes everything but the octaves equally out of tune in all keys and much is lost, in my opinion, particularly with earlier music that really uses tuning as an integral part of the music (again, in my opinion).
So. There's my summary (kind of) version of things. Still interested after all that?

Hope so. This stuff is fascinating (to me anyhow), and does bring "old music" to life in a much more invigorating way, as far as I'm concerned.
What do you think?
By the way, to "lovetheboy...", I'm glad you found my initial explanation helpful, and in regards to that book of yours - it seems you answered your own question by noting you already have found plenty of typos and errors, yes?

Keep up the questioning and research and you'll do fine, I'm sure.
JH