In my opinion, a rescue is an animal you either got from a shelter because you wanted a pet or an animal you get otherwise that you weren't looking to own but took for the good of the animal.
So as a whole, any animal you take for its own good, whether it benefits you or not.
You got the cat not because you said OMG I really want to own that cat who is nearly dead on the side of the road but because you wanted to help the cat and you chose him over a cat in a store.
Your mom took a cat that was going to end up in a shelter and then need rescuing so she just circumvented the rescue process.
Both rescues IMO. The only things that aren't "rescue animals" are animals that you take in solely for your own purposes (ie: buying a dog from a breeder or *ew* a pet store). If you get a dog from a breeder or pet store, you got it for yourself (though we all hear about "rescuing" dogs from pet stores - I don't buy into that though, you got it because you fell in love with a sickly little pup, not because you thought of the animal's interests first). If you got a dog from a shelter or off the side of the road or from a friend/neighbor/etc you were thinking of taking care of that animal because it needed help (even if you were thinking "oh nice I've been looking for a cat", you didn't go out looking for a cat purely because of the way it would affect your life).
I am willing to call legit Craigslist rehomings as rescues in most cases. I'm not talking about the person with the purebred pups who has a "rehoming fee" of $150+. Actually the cheaper the dog, the more likely you rescued it from a bad situation. People I know rescue "free to good home" horses from craigslist because we all know (even if the owner is too ignorant or doesn't care) that kill-buyers show up with wife and kids to try to say it's a good home then send them off to slaughter. I would never say that because these horses didn't get picked up by the kill buyer and then snatched up last minute in the auction pen by a legit rescue that they weren't rescue animals.
So in my view, a very broad definition of rescue is acceptable. If you picked up an animal with primary thoughts for the animal's welfare because it was in a bad situation (or about to be in one), OR any animal you pick up from a shelter are rescues.
I mean, if you want to argue that your cat wasn't a rescue because he wasn't in a shelter first... what about no-kill shelters. Is that rescue if they weren't going to die and they had nice accomodations? I any rescue is commendable and should be rewarded as such.