It's not that simple always. There might be different definitions of some of our terms depending on who you're adressing, a musician or a physicist. Most overtone singers agreed on the physical terms, as I found in this forum. But some musicians have a different approach. Compare the definitions of a tone, for instance (de/wikipedia). For musicians it can be a lot of things from sound till pitch. In physics it's a sine wave.

I would distinguish between overtone and partial, to prevent misunderstanding. Here are my definitions of terms (I hope without too much confusion with my english translation. The problems seem to be deeper discussed in German than in English):

Partials - are the pure tones (sine waves) a sound is composed of, including the fundamental. Partials can be harmonic or inharmonic. The fundamental is counted as partial no. 1.

Overtones - are the partials that are higher than the fundamental. Overtones can be harmonic or inharmonic. The second partial is counted as overtone no. 1.

Harmonics - are partials with integer multiples of the fundamental frequency. The fundamental is counted as harmonic no. 1, similar to partials. Every harmonic is a partial (but not necessarily vice versa).

Usually the numbering of partials and harmonics are similar, since in most of the cases there are either harmonic or inharmonic in most of the cases. I am not sure about sounds composed of both inharmonic and harmonic partials, as happens with some gong sounds. You'll probably agree on calling the harmonic partials 'harmonics'. But what about the numbering? It could happen that there are some inharmonic partials first and then comes a major third two octave above the fundamental. I would tend to speak of it as the 5th harmonic even if it's the 3rd partial. What do you think?

Wolfgang
www.oberton.org

Definitely

I would also call it 5th harmonic no matter what.

@ Wolfgang

Your description is right and clear, partials, overtones, harmonics, but in Italy we use only PARTIALS and HARMONICS terms.
Italian translation:
partial = componente armonica or componente inarmonica
harmonic= armonico or componente armonico

For us to say PARTIAL or HARMONIC it is the same, then we can explaine if PARTIAL is harmonic or inharmonic in relation with an audio spectrogram.

If we - in Italy - analyze a fork sound we can use both terms: