"We appoint our judges, same as every other government position. We choose our leaders, but we let them lead."

I think this is pretty similar to how it works in the US. Personally I've never heard of an election for judges. A quick google search does say:

Some states hold "retention elections" to determine if the judge should continue to serve.

So it's only certain states, and seems it's only judges who were appointed in the first place. That's a lot different than voting a judge in by election. It's essentially giving the public an opportunity to evaluate and overrule the government's selection. (I don't see what's barbaric or problematic about that.)

And, from my googling around anyways, it seems most judges don't even face these retention elections. So "Americans vote their judges in" seems maybe like loosely based on something accurate but not really accurate.

"Uhhhh... that's kind of the same thing... I might be wrong, but don't both of those mean someone who adjucates between two parties?"

I mean, whether or not I used the word correctly, the important part was my next sentence:

They don't decide guilt or innocence

They're the "referee" in terms of making sure each side follows the correct procedures and whatnot during the trial (like, "objection your honor, he can't ask that of the witness"), but then it's the jury who determines if someone is guilty or innocent.

That seems like a smaller role compared to systems where the judge decides guilt or innocence.