I'd make it a area club. Like here we have the "Mountain Valley Horse Club". You don't have to live in the area to be a member but all the club officers and most of the rides are in this area. Also when going to shows they try to take as few vehicles as possible. However, they do have a couple of members from over 50 miles away that like to come ride up here. They are not excluded because of where they live. Point being, maybe you could include the area you are from or a land mark of the area in the club name? Example "Off The Track in Mountain Valley" or something. Logos I've no idea, that's what graphics people are for. I'm sure there are a few on this forum (maybe art section) that could help out.
Riding: The art of keeping a horse between you and the ground. ~Author Unknown