A simple list of what overtones are present wouldn't tell you much. What you really want is the relative levels/intensity of each overtone. A list of the overtones with relative intensities for an instrument is called the instrument's spectrum. You might try searching for " spectrum" for the ones you are most interested in. Here's an example for a violin:
Many instruments have virtually the exact same overtones present, and often with very similar relative amplitudes. What is different about them are the relative amplitudes over time. The change in the amplitude of an overtone over time is its time envelope. The combined time envelopes of the different overtones (we'll call it just "envelope" for simplicity) is also a critical aspect of instrument recognition.
Overtones that are relatively stable of a certain period of time are called partials. Overtones that are only audible for short periods of time are usually called transients, and initial transients are important for instrument recognition.
The important aspects of pitched instrument timbre recognition are the envelope and the full overtones spectrum, including partials and transients.
Note that minor components of an instrument's sound (including noises, like breath noise) are also represented in the spectrum, mostly as transients.
Note also that unpitched instruments, like many percussion instruments (e.g., cymbals) or more "noisy" instruments have multiple frequencies present that are not harmonically related and are sometimes called inharmonic components. The frequencies and amplitudes of these less- or non-related overtones creates the spectrum for an unpitched sound and also determines whether a sound is a little bit pitched (like a snare drum). Envelopes and transients are also just as important for unpitched instrument recognition.