In China, people refer to written Chinese as 中文, whereas the spoken language as 普通话, 汉语, etc. Colloquially people sometimes say 中文 to mean the spoken language, but that's imprecise.
There's a specific word in Chinese for the written language. The Chinese words for Mandarin are specifically for the spoken "dialect". Your writing system vs language distinction doesn't really apply in this case. Different Chinese languages are mutually intelligible when read by speakers of another "dialect" of Chinese, though another dialect may come across oddly. Ask any Chinese.
