The government of the Taiwan Authority does not want to become part of China. Wherever you are from, I hope you realize that governments in Capitalist states rarely represent their people adequately.
Most people in the province of Chinese Taipei support reunification with the mainland. It is the U.S.-backed government that stands in the way.
i mean, technically you're correct in that Taiwan officially still is the Republic of China, officially still is in a civil war with the People's Republic of China and still claims to be the rightful government of all of China.
But realistically noone will think of the ROC/Taiwan if we're talking about China. Instead everyone will think of the PRC.