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I guess what people really want is the ability to say "I'm not fine" without being interrogated. In the situations mentioned above, you should be able to say, "I don't feel fine, but I don't want to talk about it right now," without feeling pressured. That way you don't have to default to "I'm fine" when you yourself are aware that that's not the case.
I don't like lying to people. I'll just tell them "don't worry about it, I'll be alright"