The post is getting super long so ask time lol i wasnt talking about teruma. I was talking purely about ma'aser. And whilst at it, as far as I remember (i could be very wrong on that though lol) nowadays every ma'aser that's given by law to levi can be given to cohen as well, since cohanim are from levi and just specific families who did the bulk if the work at the temple. But pls correct me if i'm wrong it's been a hot mi ute since I leared all of these xD
For the protocol - this is the post in question.
It's been some time since I learned it as well, I need to go over it. However, it's important to note that unlike T(e)ruma, Ma‘aser can be eaten by everyone. That's why many people actually keep it and only get rid of the T(e)ruma - due to the supposed dubiousness of Levite descent nowadays, the Halacha requires the Levite to prove they're Levite before taking the Ma‘aser.
But in relation to Cohanim: there apparently was a decree of ‘Ezra the Scribe that I can't find the source to right now that the Ma‘aser will not go to the Levites anymore because they didn't come back to Israel in great numbers during his time. This was a temporary decree, most likely, but it does show that Ma‘asrot can be given to Cohanim. There's also apparently a portion in the Gemara (Yevamot 86A) that talks directly about this, but the person saying Ma‘aser Rishon can be given to Cohanim just happens to be a Cohen himself and a 10th generation descendant of ‘Ezra, so take it with a grain of salt. Or don't, since the Gemara clearly doesn't. So in essence, you are correct.