This is a very good question, and you're on exactly the right track to be looking away from frequency. I think the place to start is with taking apart some concepts that, for convenience, most of us group together most of the time. This means defining some terms, but I'll try to keep it to the most essential.
When you ask to define "a C", you are asking to define a pitch-class, which isn't the same as a pitch. "Middle C" (aka C4) is an example of a pitch, and one octave above it (C5) is a different pitch. When we talk about pitch class, we are asserting pitches that form perfect octaves are fundamentally equivalent (which is a really well-supported, useful concept, even though in some contexts the differences between different members of the pitch-class C).
The concept of pitch-class is particularly important to the atonal, 12-tone music of the 20th century, but it is not limited to that context. Importantly, the concept of pitch-class does not depend on another concept associated with that kind of music (and musical analysis): enharmonic equivalence. Enharmonic equivalence is the principle of the piano keyboard: that C# = Db. If you are going to consider the music of someone like Webern, you would be entirely lost without using the concept of enharmonic equivalence; however, as you noted in the last sentence of your reply, enharmonic equivalence definitely does not always apply. For our broader purposes, therefore, we want to consider C# to be a different pitch-class than Db (whereas in 12-tone music, both would be part of pc 0). This means we have a lot more than 12 pitch-classes: in fact, 35 (7*5, i.e. Cbb, Cb, C, C#, C##, Dbb, etc.).
To define what a pitch-class is, then, we need to look at what makes G# (to use your example) different from Ab. Let's imagine a piece in the key of C major:
G# in that key, would be most likely as part of a secondary dominant chord (E-major) leading to the submediant chord (A-minor). Imagine a (bad, awkward) melody starting in C major that is going along and comes to ... B G# C. These notes might be harmonized by iii V/vi vi (i.e. {E G B} {E G# B} {A C E}), which could begin a modulation into the relative minor key, A minor.
Ab, on the other hand, might occur as an added seventh in a fully-diminished B chord (viiº7). Consider the same melody, but spelled as ... B Ab C. Fully-diminished chords normally occur in minor keys, so here, the harmony would likely be iii viiº7 i (i.e. {B D F Ab} {C Eb G}) and imply a modulation into C minor, the parallel key.
Even if played on a piano, where in isolation they would be indistinguishable from one another, G# and Ab convey very different musical meanings about the further direction of the piece.
By calling a pitch-class C, then, what we are doing is representing through a notation certain kinds of relationships between members of pitch-class C and members of pitch-classes G#, Ab, and so on. Any C and any G# will have the same kind of relationships to one another (of course, there are other relationships that will be specific to context.
It is impossible to make a comprehensive list of all these relationships, especially because many of them derive from the socially-constructed understanding of how Cs have been used in music throughout the centuries. Trying to understand these functional relationships is one of the major tasks of the field of music theory. One example: the relationship between C and G is very like the relationship between G and D, and we call that kind of relationship a "perfect 5th".
These relationships are very strongly related to frequency ratios and the physics of sound/overtones, but as you observed, they are not identical, for two reasons:
First, the trivial: tunings and temperaments define the ratios between frequencies, but don't specify an absolute reference frequency. For much of history, this was not at all standardized, beyond perhaps whatever the local organ happened to be tuned to. Even today, while A = 440 Hz is very prevalent, A = 415 Hz is common in performances of early music, and some orchestras are now tuning sharper (e.g. to A = 443 Hz).
Secondly, the ratios themselves are adjusted from the "pure" small-whole-number forms to fit the needs of the tuning system. Even the octave, which because it's so critical is kept in its perfect 2:1 ratio in virtually every system, can in principle be adjusted. In Equal Temperament, every ratio except the octave is adjusted away from its ideal value — yet we consider the relationship between C and G (or whatever) to be more alike than different, and the Well-Tempered Clavier to still be the same piece of music when played in Equal Temperament.
So in brief, a C (or any other pitch-class) is an abstract category which signifies that its members possess certain types of relationships to each of the other pitch-classes.
