> Both sides officially consider themselves to be China

There is no "China, the country." "China" just means, essentially, "Empire." It's like a country claiming to be Europe, or maybe better, The Roman Empire. Many States may try to make claims for the title to support their legitimacy and heavenly mandate to rule, but that doesn't make it true.

> They're the product of a civil war inside China, after all.

Only one side of that conflict still exists. The other was overthrown by the people of Taiwan in the 90s. Descendants of those overthrown maintain government positions under that party name, but it's essentially a different government, given that it's a multi party democracy now, not a single party military dictatorship.

> The current ruling party of Taiwan would like to change that, but they haven't done so for the obvious reason that the PRC would not accept it (and most Taiwanese people prefer to just leave things as they are).

This is mostly true, with caveats: Most people in Taiwan prefer independence, but don't want to declare it to trigger a war, so therefore they only prefer status quo because it involves independence without war. If they could get it, most Taiwanese would prefer declared independence with no threat of war, but pragmatism rules out.

I'm also not sure I agree the DPP is necessarily pro-overt independence, just the current president tends to use more aggressive language than normal.