I don't understand. Is Taiwan or isn't Taiwan an independent country? If it isn't, then did we just declare war on China?
Diplomatically? No, most countries officially recognize it as part of the PRC.
Why the inconsistency? If you recognize Taiwan’s independence, the PRC won’t talk to you. If Taiwan declares independence, the PRC will invade (or at least has threatened to).
Pretending that Taiwan is part of China while also making it very clear that we will defend Taiwan from China bit silly, but being silly is better than having a great power war.
This isn't something that can be understood outside of the "ends-justify-the-means", "but my pragmatism" and other sausage making slogans of the foreign policy establishment. We'd have to go through all of the historical machinations before we'd even begin to have a context.
You're not wrong for looking for some semblance of logical consistency. If you prefer analysis from first principles, that's fine, I'd even agree with that perspective in many ways, but you'll have difficulty evaluating the status quo of geopolitics.
Maybe a long way of saying that it doesn't make sense, it isn't supposed to make sense under scrutiny, it probably never will make sense and the expectation is for knights-errant.
But it is, obviously. Anyone who's not a moron can see that.
Fast forward to sixty years after the fighting just stopped one day. How would you view those two political entities? Are they equal? Does the existence of the CSA mean the USA doesn't exist as a country? What if the CSA implements trade policies which dissuade you from recognizing the USA as an independent nation? Does the USA still exist in your eyes? Is the USA or the CSA the "true" country?
The answer at this point in time is quite simple: the Republic of China (aka Taiwan) is clearly an independent country. However, the People's Republic of China (aka China) holds such economic power, they bully the rest of the world (though trade policies, etc.) into not recognizing Taiwan as an independent country. Also, the combination of geography and maritime law means that if Taiwan is an independent country, China has very little direct access to the open seas. They want freedom of navigation, so there's a geopolitical angle as well. Look into the construction of artificial islands in the South China Sea to advance the Nine Dashed Line policy.
You also need to remember that Chinese civilization is extremely old. The PRC claims to hold something called the Mandate of Heaven, which is a fairly important concept in Chinese history, as it is divine authority to rule over the Chinese people. Frankly, based on how the PRC acts, the RC clearly holds the Mandate of Heaven these days.
The Mandate of Heaven is rather more like a justification of de-facto rule and claiming to have lost it is an argument against a failing dynasty. The dynasty managing to unite China would declare it, and as it eventually collapses, rebels would declare it to be forfeit. There would be messy civil war, one of the warring groups would dominate all the others, and the cycle starts anew.
This is totally fine for commercial traffic. Since China has a huge export market, it is also not interested in raising shipping costs.
In practice, it is an independent country. But the People's Republic of China (PRC) has a "one China" policy where they claim Taiwan as theirs.
PRC spends a large deal of their diplomatic efforts trying to enforce their policy.
The short history version is basically like this: after the overthrowing of the last emperor and WW2, the Chinese civil war continued between the KMT political party and the Communist party. KMT basically lost and moved to which is Taiwan today and even the KMT now is no longer the controlling majority now (they have elections whereas mainland China does not).
Though, in my experience, asking that question to people in China (or from there) is generally met with:
* I don’t know, not interested in politics
* Chinese people live there so they’re a part of China (not separating gov from race/ethnicity as a naive answer)
* (no comment)The Taiwan situation has been escalating for awhile now that almost everyone expects an invasion before Xi’s lifetime term ends.
China just uses its weight to bully countries into submission on declaring their view on Taiwan.
Taiwan is a sovereign nation in most respects, at least to the extent that any non-superpower is. China insists stringently on the legal fiction that it is a part of China controlled by a different faction in a frozen-but-ongoing civil war, and other countries - not least Taiwan itself - go along with that legal fiction under the implicit or explicit threat of Chinese force; a threat that is less effective against the US than against many other countries.
Is this an act of war by the US against the PRC? Eh, maybe, if you accept the PRC's territorial claims. A lot of technically-acts-of-war happen, especially in situations like this. Again this is a case where the abstraction breaks down and the realpolitik becomes paramount.
However, US actions have also been clear. In 1996, the US sailed a carrier battle group right down the straits when things get testy. The US has been selling weapons to Taiwan for decades. The US has enacted export limitations against one country, but not another.
Functionally, this action did not declare war on China. Yes, China leadership could find a way to stretch this into "war", but they also could have stretched any of the other things the US has recently done (including Biden straight up saying that the US would intervene if China invaded).
That being said, credibility is important when making threats. And letting your lines in the proverbial sand get crossed, moved, erased, redrawn, etc. endlessly can make for some ... interesting geopolitical situations.
The presence of US troops means that if China attacks, the USA is going to suffer military casualties, so there is almost certainly going to be war between the CCP and the US.
Without US troop presence, things are more ambiguous.
The trainers turned themselves over to the Russians, and were interred (at the Ritz) as neutral soldiers.
A training team can be a tripwire, or not.
Again, ambiguity.