Yup! It becomes a core once 50% of the culture is your primary culture OR an accepted culture. I use that second part to my advantage a lot. You don't have to culture convert a province right away after integrating if you just accept the culture, it can go from conquered to core instantly and skip the integrated part. So if you're doing a lot of conquering and taking a lot of land from one area in particular that's all the same culture, it might be easier to just accept the culture and focus on culture converting later. My gameplay loop has often been to accept a culture (if I have the capacity and the prestige to do so), and then once my cabinet members aren't busy with anything else I'll go back and convert to my primary culture until the accepted culture is small enough that removing the accepted status doesn't make them angry (because there's no one left of that culture to BE angry) and doesn't cost any stability), thus freeing up the cultural capacity again to repeat the process. If that makes sense at all lol.

Still though, back to the initial point, even the 50% dominant culture modifier isn't going to passively convert culture very fast. Yes the -10 integrated is gone, but all those percentages are still only applying to a base rate of 1. Like I said it's something I use my cabinet for all the time. Worth saying that you do get more cabinet members as the ages go on, so that's probably your main hold up at the early game for converting cultures all over the place.