Edit: I never realized the fundamental was a harmonic. I thought all harmonics had to be above the fundamental. I thought the overtones were the multiples (e.g. higher octaves) and harmonics were other intervals produced by the instrument (organs being the most extreme example and maybe oboe being the least)
Even though the terminology can be defined in order to make these distinctions clear, there's an arbitrariness in terms of which definition you prefer (in the past, some mathematicians treated 1 as prime, which makes the definition of a prime simpler, but makes many theorems about primes more complicated to express). And there's a likelihood that even experts will sometimes use the simple term informally when they technically mean a more specific thing (like occasionally saying "divisors" instead of "proper divisors", or something).
I imagine that acoustics experts have a definition available that definitively states whether the fundamental "is" a harmonic, but in certain contexts it intuitively makes sense either to include or exclude it, regardless of that.
A similar case might be "animals"; taxonomically humans are animals (and apes), which is very important sometimes and confusing other times.
Overtones are any partials other than the fundamental. While overtones CAN be non-harmonic (notably in bells), I think it's fair to say that most of them, or at least the most important ones, are usually harmonics. This is because the terms "fundamental" and "overtone" are historically music terms, and are generally applied to sounds we perceive as tonal or musical: and such sounds are largely composed of harmonics.
Not at all! It's the other way round. Harmonics are those overtones that are integer multiples of the fundamental. You can have other, non-harmonic, overtones.
Now what happens when I play a chord and get overtones out of the interaction between two strings or a choir? What's happening there?
Nothing. Sound superposition is linear. There's no interference between different frequencies. By playing several strings together you obtain the sum of the sounds played by each of them separately. No new frequencies can appear.
The beats have a frequency and are audible, but their frequency is not a pitch.
You can deconstruct any given sound wave into its partial sine waves. This is typically done using a Fourier Transform algorithm (FT).
Of course, we can also do it the other way around, create a signal out of many sine waves. In practice, this is called additive synthesis.
Instruments not only sound different because of the timbre. Another thing to look at is the loudness over time, e.g. when plucking a string. We usually do this with an ADSR representation (Attack, Decay, Sustain, Release).
Knowing these chracteristics is enough to recreate an instrument with a synthesizer. Of course, it gets more complex when there are inharmonics and if they are irregular (different depending on each tone). That's why synthesizing instruments realistically is a pretty time consuming science.
Since you mention "octave" here I want to point out that this is a common misconception. Harmonics are not just octaves of the fundamental. The octave scale is logarithmic but harmonics are linear and include all integer multiples.
If your fundamental is 100, the octaves are 200, 400, 800, 1600, .... But the harmonics are 200, 300, 400, 500, 600, ... There are many extra harmonics that aren't overtones.
This is important for many reasons, but a fun one is that you can use this in sound design by relying on a clever thing our brains can do. We are so good at doing frequency analysis in our heads that we can figure out what fundamental must be present even when it isn't. If you play sine waves at 200, 300, 400, 500, 600, etc. your brain can figure out that those would all be multiples of a 100-Hz fundamental, even though that fundamental isn't present [1].
This lets you do a neat trick where you hi-pass a sound to remove some of the lowest frequencies in order to make room in the mix for other bass sounds. Even though the sound loses its fundamental, listeners will still hear it as "functionally" having a bass register. (This is also why when you listen to music on a crappy tiny speaker, you still hear the bass as bass even though it's actually quite tinny and high-frequency.)