What is the word used by people in Taiwan to refer to "the Chinese language"?
The official language in Taiwan is standardized by the government of ROC (Republic of China, 1912 - present). The official language is called 國語 and the written form (or written language) is 國文.
The English word for 國語 (The National Language) is Mandarin but that word has a much longer history than 國語. See wiki mandarin
(MANDARIN has nothing to do with 满清, 满州國, 满大人, whatsoever)
A language is primarily a historical and ethnic matter. The Chinese language would be more objectively called 漢語 or 汉语 (Han Languages) which is a group of languages or dialects. 國語 (普通话) is just a standardized (or normalized) dialect, no more, no less.
Is 中文 the most "neutral" term for the name of the Chinese language? Not really. It depends on what you mean by the word 中文.
(1) 中文 is an abbreviation of 中华语言文字 (中华, 华夏, 汉民族)
(2) 中文 is an abbreviation of 中国语言文字 (中, 中原, excluding 四夷,BC 1000)
(3) 中文 is an abbreviation of 中国语言文字 (中, 中国, 中华人民共和国 1949, PRC, not ROC)