"I don't mind" vs. "I don't care"

Are these the same thing or is there some subtle difference that I'm missing? To me "I don't mind" is what NTs say when they don't care, whereas I just say it like it is. People always seem slightly taken aback when I say I don't care, but it's honest—maybe too honest.

Examples: What would you like for dinner? Which shops would you like to browse in? When would you like to book a holiday? How would you like the flower girls to have their hair at our wedding?

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