The only chord notation I find in any of my refs that uses the "1/2" fraction is in The Encylopedia of Picture Chords, compiled by Leonard Vogler, Amsco Publications ©1990.

A table of alternate notations at the back of the book gives:

1/2dim; 1/2dim7; m7(b5); m7(-5) as alternate notations for the chord the book notates as a ^Ø7.

(For those whose font tables may be different than mine, the "raised" character just in front of the last "7" resembles a miniature "phi" or "o-slash.")

The book doesn't give a "name" for the ^Ø7 chord that I can find.

This doesn't appear to be anything likely to be indicated by the marking in question.

I'll note that this book doesn't illustrate a barre form for any of the other chords I saw in a brief scan through.

The "half-barre" suggestion sounds most reasonable. Does a "half-chord" (the literal reading if it is a CH and not a CII) have any meaning to anyone?

John