So you're looking for objective data to force the warranty company to install a higher quality unit. Good lucky. Home warranties aren't worth much more than the paper they are printed on... or rather seem to be more of a hassle than they are worth and you'll end up spending a lot of money just to "fix" what they supposedly repaired.
It's too bad the warranty companies won't just cut you a check based on a low bid quote for what the repair costs.... then you could apply that towards a good quality repair or installation from a reputable company.
Form the warranty company standpoint, the Payne POS builder grade... possibly used unit is 13SEER, XX tons so therefore it's comparable.
However, unless they changed the indoor coil at the same time, it's possible that the A/C system is no longer 13 SEER.
I don't know why they couldn't just hire a local trane company to fix the existing unit. Sounds like it just needed a couple new parts.