It is part of my education/formation and job to know about harmonic decomposition of waves (whichever you want, pick your poison... let's talk about music, so... acoustic waves then).

Harmonic decomposition of waves comes from Fourier signal analysis, a mathematical tool which asserts that any periodic signal (which implies a frequency of oscillation/repetition), regardless of how complex it is, can be decomposed into a sum of simple periodic signals (sine waves). It also establishes that, in the case of musical notes, there is a lowest frequency, which is called fundamental - or first harmonic, since it is the first sine wave in the sum decomposition - and that all the other harmonics (the other sine waves in the sum) have a frequency which is a multiple of that fundamental frequency. Each sine wave in the sum has a proper 'contribution', i.e. an individual amplitude which depends on (and can be computed from) the whole signal/note.

Whether one harmonic has an amplitude of 0 (the corresponding frequency being actually not represented in the whole signal/note) is only relevant for the application of the theory/method to real cases. Hence, the numeration of harmonics is only dependent on the frequency it represents.

It is my understanding (but I may be wrong here) that the term 'overtone' comes from music, which is a different approach for the same objects : harmonics. The approach is different in the sense that a musician is only interested in what he/she can hear. There is no point in acknowledging the theoretical existence of an harmonic which cannot be heard, thus overtones numeration skips the absent harmonics. I may be wrong about etymology but it seems to me that the very term 'overtone' implies 'a tone over [another one: the fundamental]', and as such the fundamental cannot be an overtone.

@randulo :

1) I didn't read all the page of your link but for what I have seen there is no contradiction with what you said previously in your post,

2) I think somebody nailed it when he said : you count up from the lowest note you can produce, and you count from 1 onward. So, the lowest C you can play would be C1 regardless of the sax you're playing.

Edit: 'somebody' was in this case

@Nick Wyver...

Edit(2): lowest C (on the sax, not concert)