Formal correctness is overrated. But yeah, it's correct because the word "belief" has multiple meanings (it can be a conviction or confidence or faith), and believing in someone implies that you have faith in them, whereas believing in, say, single-payer healthcare implies you have a practical conviction or opinion.

But there's really no difference in my opinion. When someone's about to play an important football match and you tell them you "believe in them", you're really saying you're convinced of their ability to win or otherwise perform well.

Hero Bash: If I'm understanding correctly (which I may not be) you're wondering whether or not both meanings of "believe" are correct? As in, they are both accurate? If so, yes they are.

I think he was mostly wondering whether it's correct that the meaning of a word can be modified based on what the other words in the sentence are. And yeah, it can, and it's correct. Words have multiple definitions and which definition is right depends on the context. It's like that in every language.