Mainland China refers to China sans Hong Kong and Macao, yes. Because you have to distinguish and "Mainland China" is a whole lot less typing than "China but without Hong Kong and, oh yeah, Macao too".
The official name of Taiwan's government is Republic of China, and it's the continuation of the government that controlled all of China before Mao's uprising.
So I'm not sure your argument makes sense. They claim to be the legitimate Chinese government in their own name.
Outside of that very specific context, China == PRC. I understand the history of Taiwan very well, and it's important for people to understand that they are two different countries.
No, that's just completely wrong. Taiwan's government was created when Chiang Kai-Shek fled mainland China in the 1950s. It's the continuation of the Republic of China that governed the entire country of China before the PRC government took over.
Saying that the PRC created the ROC name is ridiculous. The ROC predates the creation of the PRC and is the older name. The PRC didn't even exist when it was created. Also suggesting that the PRC would ever endorse the ROC in any format is silly. Their entire position is that the ROC doesn't exist.
You should go read a Wikipedia article on the history of Taiwan as your grasp of the history here is not strong.
Edit: People downvoting this have absolutely no clue at all about Taiwanese politics and don't even bother googling "Taiwan independence". If they did, they'd knew that Taiwan never formally declared independence.
Show me where. Taiwan claims to BE China, not part of China. That's a big difference.
Also, they don't need to claim independence since Taiwan has never been part of PRC. That would make as much sense as France never claims independence from the USA.
And yet there’s a whole wikipedia page saying the opposite?
No, not really.
The Taiwan independence movement is a political movement which advocates the formal declaration of an independent and sovereign Taiwanese state, as opposed to Chinese unification or the status quo in Cross-Strait relations.
Currently, Taiwan's political status is ambiguous.
Former president Lee Teng-hui has stated that he never pursued Taiwanese independence. Lee views Taiwan as already an independent state, and that the call for "Taiwanese independence" could even confuse the international community by implying that Taiwan once viewed itself as part of China. From this perspective, Taiwan is independent even if it remains unable to enter the UN.
Most Taiwanese and political parties of the ROC support the status quo, and recognize that this is de facto independence through sovereign self-rule. Even among those who believe Taiwan is and should remain independent, the threat of war from PRC softens their approach, and they tend to support maintaining the status quo rather than pursuing an ideological path that could result in war with the PRC.
The questions of independence and the island's relationship to mainland China are complex and inspire very strong emotions among Taiwanese people. There are some who continue to maintain the KMT's position, which states that the ROC is the sole legitimate government for all of China (of which they consider Taiwan to be a part), and that the aim of the government should be eventual unification of the mainland and Taiwan under the rule of the ROC. Some argue that Taiwan has been, and should continue to be, completely independent from China and should become a Taiwanese state with a distinct name.
On 25 October 2004, in Beijing, the U.S. Secretary of State Colin Powell said Taiwan is "not sovereign," provoking strong comments from both the Pan-Green and Pan-Blue coalitions – but for very different reasons. From the DPP's side, President Chen declared that "Taiwan is definitely a sovereign, independent country, a great country that absolutely does not belong to the People's Republic of China". The TSU (Taiwan Solidarity Union) criticized Powell, and questioned why the US sold weapons to Taiwan if it were not a sovereign state. From the KMT, then Chairman Ma Ying-jeou announced, "the Republic of China has been a sovereign state ever since it was formed [in 1912]."