Believing in Christianity while practicing the 613 commandments would be not ok.
I always wonder about that. I know it’s true, but I am not sure what argument they put forward to explain that. I can’t remember anything that requires one to think anything specific in the Tanach.
I believe you may be right for Christianity, but not for Judaism, for instance. For Judaism, I think, it is not very relevant why you act righteously, as long as you do. As long as you don’t eat pork, worship God, don’t light a fire during Shabbat, and do the other 600+ things, you are probably OK. Although I am not a practicing Jew, so someone may correct me.