Well, now we're gonna get nitpicky :) Unicode is plain text, therefore emoji are in the set qualified as "plain text." "ASCII text" may be more accurate.
I'd say your first description, "text-obsessive", was the most accurate.
"Plain text" is a technical term, which generally means raw unicode (or some other character encoding) without additional markup. Emojis are unicode, and can exist in plain text files.
"Text" does not necessarily mean unicode, nor any type of character encoding. The books on my bookshelf are filled with text. If any of those books contain emoji, I would consider the emoji to be an illustration, not text.
(This is a silly discussion but also the sort of thing I have fun thinking about.)
