I was asked by a friend why I use 中文 (zhōng wén ) instead of 汉语(hàn yǔ) when I refer to Chinese language. Well, 汉语(hàn yǔ), meaning Language of Han, may reflect “language” better; but it limits itself to only language. When teaching kids Chinese in a non-Chinese environment, teaching the language is only part of the task; introducing Chinese culture and customs is another inseparable part. In this sense, 中文 (zhōng wén ), Chinese Literature, seems to be more appropriate.
Moreover, Mandarin Chinese is indeed based on the language of Han people. But given 56 peoples in China, using 汉语(hàn yǔ) to refer to the whole Chinese language seems to me not comprehensive. I would rather use 中文 (zhōng wén ).