Wow, didn't know this was so confusing. Backstraps are straps of meat that run down a deer's back. That may not be the scientific term for it, but when you're talking meat to a deer hunter, 90% of them are going to agree with this.
I've heard them both referred to as tenderloins, as the backstraps are the tenderloins on the outside, and the inner loins, or "fish", are the tenderloins on the inside. I don't know if this is correct, I just refer to the straps of meat on the back as "backstraps", and the loins on the inside as "inner loins", and I've never had anyone not know what I was talking about.
Backstraps are where your chops come from, so if you want your backstraps as steaks, don't get chops. Depending on how many deer I get per season, I do both. If I don't get at least three deer, I do chops to spread it out more. That's enough to keep me in the woods until I get my third and fourth deer.
The inner loin is, IMO, a better cut of meat, but less popular because its so much smaller (and apparantly disappears from some butcher shops). One inner loin is one serving, where one backstrap could be four or five servings, (depending on the size of your plate). Both are excellent cuts of meat.
It seems to me that if your butcher hands you a bag of ligaments when you ask for your backstraps, he might be a bit of an @$$. If your butcher doesn't know what you're referring to when you say "backstrap", find another butcher.