Try having a look at this thread from a few months ago
It may not look too relevant to start with, but thread drift moves it into the right territory after a few posts.
Whether you can log your short check-out flight depends on what capacity you operated the aircraft during that flight, rather than were you a group member or mentioned on the insurance.
I'm a bit confused about the group members/owners who all seem to be able to give/sign-off differences/tailwheel training, but aren't FI's.
Are they CRI's then? If they are, then yes, they can carry out your differences training on your group aircraft and you can log it P.u/t.