There's a bit of confusion in terminology. You don't play on frets, you play between them. So you actually "play a fret" by placing your finger behind it.
Ex: The fret marker (dot) at the 5th fret is actually between the 4th and 5th fret. This is where you place your finger to "play the 5th fret".
In Griff's diagrams he is telling you which "fret" to play. He isn't pointing out the actual number of the fret.
In his A7 diagram back on page 13 he has 5fr, meaning that your finger is actually between the 4th and 5th fret. He isn't referring to the actual fret next to 5fr ,which would be the 4th; i.e. he's not labeling the fret, he's telling you which fret you play behind, which would be the next fret, the 5th.
Following his convention, the C9 on page 16 is played with you first finger behind the 7th fret, between the 6th and 7th.
You are correct that if he was numbering the frets themselves it would say 6th fret where it says 7th, but he's telling you to play the 7th "fret" not labeling the frets themselves.
Notice in your screen-shot above that his first finger is directly on the 7th fret dot. This is where we play the "7th fret" even though it is actually between the 6th and 7th fret.
It's just a matter of convention in chord diagrams. Sometimes authors place a number next to an actual fret they are referring to and sometimes, like here, they refer to the "space" between the actual frets; i.e. the position where you place your first finger.
