Is paying when you're under threat of credit damage really admitting fault?

In many cases, yes, part of the settlement is an agreement on liability. It's not required, so yours may not have, or you may have specifically declined liability.

Regardless, though, you will have difficulty suing them for taking your payment when you agreed to pay it. They said "We think you owe us $1600" and you said "Here you go!". You decided on your own that a payment of $1600 was better than the other option. That's the whole point of a settlement. You negotiated an agreement that you were worth paying.

Now you have regrets. That's basically a "tough shit" moment. Not only is it economically infeasible(you'll likely spend more to press it than you'll get back) but it's also a grand waste of time. The time to fight this was then, not now; and the company can easily win by saying "Hey, we made this agreement to pay, and we've upheld our end, he should uphold his."

Finally, your chances of prevailing in any suit are pretty small. Your entire narrative reeks of someone who's trying to get out of an accident he caused. Regardless of your actual fault, you have to admit that this story stinks to high heaven. Every car company tells you to report damage to them before you leave the lot. You noticed it, didn't put it on the paperwork, and didn't tell anyone when you left. You also didn't follow up. Then, when you returned it and claimed it was pre-existing, you didn't follow up with the manager like you were told to do.

Your narrative basically insists upon everyone at the rental company who saw this car - and I mean everyone, from the guy who checked in the person before you, to the guy who washed the car, to the guy who prepped it for you, to the person checking you out - just totally missed this obvious damage. And you, as well, completely forgot about your responsibilities until you happened to get called out on it. Then suddenly you remember that it's been there all along, although you told no one.

And maybe that is what happened. But it's far, far less likely than you having done the damage and trying to weasel out of it. And when you go ahead and pay the bill rather than, say, disputing a credit hit later, you look like you're rolling over and admitting guilt.