I'm not asking about obvious cases like regular 6th chords or extended chords with 13ths. For example, I'm clear that C E G A is a C6 chord. I'm also clear that C E G B♭ D F A is a C13 chord. (And yes, I know that, in practice, some of those notes—the 5th, 9th, or the 11th—may be omitted.)
I'm interested in theoretical and edge cases. Let's begin with an example like C E G A B♭. Wikipedia says it's a C7/6 chord. Fine.
The 7/6 notation seems to begin and end with sevenths, however. There is no 9/6 chord that I can find, for example. (That's not to be confused with a 6/9 chord, which does not have a 7th.) So, if we had C E G A B♭ D, what would that be, assuming C is the root and we wanted to leave nothing to ambiguity? Our choices are:
Again, I'm sure someone's going to say that you can omit the 11th from a 13th chord, so C13 is appropriate. But what if I wanted to find a chord symbol that was unambiguous about what's in and what's not, irrespective of the voicing?
If we root the chord on G, we have a similar problem:
(I thought of Am11(♭9), which is perhaps the simplest of all, but that's not useful to the question.)
What plays into the decision to call it a 6th or a 13th? Sure, we could say it's the proximity of the 6th to the root, but that doesn't feel totally comfortable to me because chord symbols don't necessarily imply or reflect chord voicings. So what is the rule?
I understand the answers I'm getting. Thank you. I was hoping for answers which don't take context into consideration, though. (Think of an algorithm which generates possible chord names given nothing but a set of pitch classes.)