I have also read the 1/3 rule for mortise and tenon joints, although I would be pressed to find the reference.
If the tenon was going to be a tight fit, then once glued in place, either 1/4in or 3/8in would work, since the tenon would be attached to the surrounding wood of the mortise, so the joint would be strong.
I would not use a 3/8in loose tenon in 3/4in stock, I would be concerned at the thin 3/16in wall of the mortise. This can be overcome by using a gap filling glue like epoxy.
Personally I prefer to use the yellow glues where possible. Easier for me to deal with.
I would therefore use a 1/4in loose tenon.