Well, to double check and make sure that you post up F/C instead of F is a start ... as for the F/C on the 8th fret... if there's not too many mistakes during formatting, you could make a note at the bottom of the song indicating that it is played on the 1st fret not the 8th as the chart shows.
Otherwise, it seems you'd have to make up your own chord chart drawing, upload somewhere on the internet, and then link to it. If there's a bunch of chords that you do that with regularly, I would say that it would be safe to put a link to such a photograph in your signature and note something in your signature about referring to the link for the chords played in your songs.
Now... on the other hand, I just looked at the large chord chart (because I too play the F/C on the first fret) and there IS one in there if you go searching for it. Once you find it, you can do the same thing with it that you would do with any other picture to post it somewhere else. (right click, choose copy picture url, come back here and paste it between img tags -- see this post for more details on how to http://www.chordie.com/forum/viewtopic.php?id=14150)
So you could reasonably do that at the bottom of your song posts... something like:
Please note, chords used in this song are actually:
F/C:
yada yada yada ... I think if you put a TON of those in your signature however, it would be a bit much (I'm not sure that we have rules against pictures in siggys, but they might be blocked since no one has any anyhow). OR... you could make a post JUST of the chords you use in your songs, and then refer to it at the bottom of each song, or in your signature.
I hope I've given you a couple suggestions that make sense and will help you figure out what will work easiest for you. I know you post up TONS of songs on here, so easiest would be best I'm sure rather than having to work even harder coding extra stuff into each and every one. Good luck!
Amy