From what I've always understood anything such as G/F or C/G, the letter after the slash is a reference to the bass note to be played, although after 40 years on the guitar I can't say I've ever seen a reference to a G7th as G/F (although F is the 7th note in the chord). One of the more common inversions that I've seen laid out is the root / followed by the third which would be played an octave lower than the norm.
Just my 2 cents worth.

------------------
...shboom