Invaded Taiwan? Oh jeez. It was a civil war. And the KMT lost and fled to Taiwan.
Separate countries? All but ~10 countries on Earth including the US and all of Europe have what's come to be called the One China Policy. Taiwan is not recognized as an independent nation. It's really Schrodinger's nation, both independent and not at the same time and God forbid anybody opens the box and forces the issue.
The one China policy, which is the US stance and distinct from China's "one China principle", is accepted by the US with a wink fingers crossed behind their back.
De facto vs. de jure is what you're trying to describe, and I'm sticking with the practical one.