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You can definitely say "read Mandarin", because particularly in informal contexts there is definitely a difference between "Standard Chinese" based on Mandarin and, say, spoken Cantonese written down.

You can definitely write Cantonese so I'm not even convinced you are correct.

You are partly correct. Mandarin usually refers to the spoken language and Chinese to the written characters.

However there are still differences in writing / vocabulary even if you disregard the existence of simplified and traditional versions. It's similar to US and British English. Books are are often translated to both versions and it's similar for (Mainland) Chinese, Hong Kong Chinese or Taiwanese Chinese text with even more extreme differences.