"American Chinese" means "American-born Chinese".
"Chinese American" means "Chinese-born American".
Are my understandings correct?
Please note my alterations above. You need to mark out words you're asking about in some way. I have put quotation marks round them. You need the hyphens as shown above.
I don't necessarily agree with your understanding. For me, if someone describes themselves as "an American Chinese", it could mean that they hold both a US and a Chinese passport, or that they are of Chinese heritage but US nationality (passport), or that they are of American heritage but Chinese nationality (passport). It could even mean that they have an American mother and a Chinese father (or vice versa) but, for all we know, they could have been born in Holland and have Dutch nationality. In both cases, if you wanted to know for certain what the speaker meant, you would have to ask follow-up questions.
Neither of them says anything about where they live (I know you didn't suggest that they do but I just thought I'd point that out).