For me, if any money changes hands, you aren't rescuing that horse - you're buying it. If money was in play for a horse in terrible condition, you "upgraded" that horse.
Really though, I think the word "rescue" has been so overused that it really has no meaning anymore. It's becoming a cliche that any Joe Schmoe uses to justify that skinny horse in the backyard or why their mare/stallion must breed - "Well, I can't ride it. It was an abused rescue
It's becoming a glamor word that people drop to justify the ill mannered behavior their animals exhibit and to get congrats from people they run with.
If I wanted to, I suppose I could say that Lacey was a "rescue." I was basically paid to take her, she was a week away from her date with the needle, and she had some serious mental demons. HOWEVER, to my mind, she was not a rescue because her previous owners adored her (however misguidedly), she was not skinny, and she wasn't lacking in hoofcare. She was massively upgraded and I don't think she'd choose to go back there if she were a human, but she wasn't out and out suffering.
At the same time, I do think of my cat as a rescue since I found him abandoned in the woods as a near death 5 week old kitten. I had to spend quite a bit to get him healthy and if I hadn't pulled him out, he would have starved/been eaten by something in those woods within the next 24 hours. Even the vet was pretty sure he wasn't going to survive but he totally did.
Personally, it's not a huge deal to me but I basically refuse to congratulate someone who describes their animal as a rescue. I figure that the people who really do rescue animals are the ones keeping quiet about it because they don't see it as a big deal - they're just doing their "duty." I don't go around saying "This in my RESCUE cat, Booth!". I tell his story if people are interested and let them make their own conclusions.
If people want to try to force themselves to be "cool" via supposed animal rescuing, they can be my guest. I just won't have very much patience for them to regale me with stories about poor poor Cowbell, who was so abused an traumatized when they rescued her for $500. Poor baby. Yeah, NO.
I also think pitying an animal for the situation it came out of is rather silly. Animals live in the now - they're focused on the good treatment they get now, humans are just funny and don't think that way..