The definition of a "country"/"state"/"province" is not agreed upon and usually politically motivated. In fact, there is a very informative CGP Grey video on this very subject (including Taiwan's status). You might claim that he's an "American", but I think his presentation regarding the actual definitions for a country is completely reasonable.
That said, even though the actual definition of a "country" is somewhat of a political game, there is an actual need to define Taiwan's status for practical purposes. The (conservative) Kuomintang has always claimed that it is the sole legitimate government of "China". However, I think practically speaking, nobody would agree (even Taiwanese citizens, both anecdotally and through surveys) that Taiwan has any political claim over the government of the mainland. Similarly, the opposite applies, although not economically. This is what we commonly mean when we say Taiwan is not a province of China. In fact, even the CCP doesn't claim that Taiwan is a province under their jurisdiction. In fact, one of the prevailing political issues in Taiwan today is whether or not to pursue independence from China.
This might look like me dodging the issue by redefining what I mean, but I think it's some much needed clarification. I hold that we're debating this issue because we care about relationships between countries (and the economic/travel implications thereof), and I think a practical definition is much more useful for this purpose. Feel free to disagree. If you do, I don't have an opinion on it, and I'll assume you're completely correct regarding the semantic definition of a "country".
This has been an interesting exchange. It is clear that you are not going to change your mind, and that's fine. I also think it's fair that any future readers (?) will be able to make up their own minds regarding the subject, since you have kindly linked Taiwan's constitution. There's not much more I have to add on the subject, but feel free to add points you feel have been overlooked. Have a nice day! :)