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I sometimes start with a password chosen by other means, then I flip a coin for each character to decide if it is lower or uppercase.
This seems like the worst solution of all. You've just added a bunch of shift keypresses and a ton of difficulty remembering what letters are uppercase for very little extra entropy. Each coin flip is just one bit.

Part of the thesis of the post is you can ignore uppercase characters and symbols altogether. You'd be better off just adding one diceware word from the short list or two random lowercase characters. Easier to remember, less typing, and better security.