Dodgers catchers' hit tools are where things get interesting
Again, we can operate safely under the assumption that Cartaya's offensive and defensive upsides are higher than Rushing's, as his physical tools are pretty ridiculous. However, the most important tool a guy can have, the hit tool, is where the argument for Rushing becomes more pronounced.
In the lower levels of the minors, Cartaya was aided by the fact that worse/less experienced pitchers struggle to throw strikes and were also probably a bit wary of his reputation. His strikeout rate was pretty consistently around 27% and he was drawing walks at around a 13-14% clip. However, now he is at Double-A, and while it is still early and a small sample, his strikeout rate has jumped up to almost 33%, while his walk rate has plummeted to 6.3%. As a result, he is slashing .186/.253/.314 in 18 games so far this season with just two homers.
Rushing, meanwhile, has performed much better, albeit against a lower level of competition in High-A. There, he has put up a .284/.462/.602 line with seven homers in his first 25 games of work with significantly better strikeout and walk rates. Some of that can be chalked up to competition level and small sample size weirdness, but Rushing's hit tool has consistently been graded higher than Cartaya's to begin with. He is selective at the plate, has better bat to ball skills, and is unafraid to draw a walk.
The end result has been, at least so far, that Rushing has been able to get to his power more in games because he is better at making contact. Between that and being a college draftee, he is on the fast track to joining Cartaya and getting promoted to Double-A himself, which will make things even more interesting.
In the end, there is still plenty of time for both Dodgers catchers
The Dodgers are lucky that they don't have to make this decision now. Will Smith is both awesome and under contract through at least the 2025 season, so the team has time to let both of these guys develop. Despite his struggles to start this season, Cartaya has enormous upside. The team clearly loves having him in the organization (trading Keibert Ruiz to allow him to grow) and are taking great care to make sure he develops properly at a position that takes a lot of time to learn.
Rushing, meanwhile, has been a very pleasant surprise as a second-round pick that has turned into a significant catching prospect who, as of this moment, may actually be closer to big-league ready than Cartaya right now, given the current hit tool difference. He can hit and hit for power, and there isn't anything defensively at the moment to make one think he couldn't catch in the big leagues.
There are reasonable arguments on both sides of this debate at the moment as to which prospect one prefers. Given enough time, these sorts of things sort themselves out. One thing is absolutely true, though: the Dodgers' minor league catching depth is in a really, really good place.