Yaaaaay! Woot Woot.
Ding Ding Ding - We have a winner lol.
Yup - Anytime a 13 is noted (Say C13), a dominant base chord voicing is automatically assumed. In the instance of a 6 chord (C6) it is not.
There are 2 types of voicings in the anatomy of a chord. The base (not bass) voicing, and the upper structures. The base voices are 1 through 7, and upper structures are 9 through 11
If the chord does NOT contain a dominant voicing in a bass, which by definition is a major 3rd and a flat 7th, the 13th chord BY DEFINITION is not a 13th.
If the 3rd is raised and the 7th is also raised, it is a Cmaj13
If the 3rd is lowered and the seventh is lowered it is a Cmin13
If the 3rd is raised and the seventh is lowered it is a C13
If the 3rd is lowered and the 7th is raised it is a very ugly chord that doesn’t really have a name (but it would not be a 13 chord, since it does not meet the criteria of a 13 chord by definition).
This parts technically incorrect, but you seem to have understood the other part anyway.
Just to clarify, C is indeed the ‘dominant’ chord in F like you said. However there’s a difference between a dominant chord, and a dominant chord voicing.
If you are in the key of C, then F is your subdominant chord. Lets say we play an F7 in the key of C. Then we are playing a dominant voicing, on a subdominant chord.
So the 13th voicing doesn’t have anything to do with the Tonic, Superonic, Mediant, Subdominant, DOMINANT, and Subtonic, and Tonic Leading (Degrees 1 2 3 4 5 6 7 respectively).
It only has to do with the intervals within the base voicing of the chord