ELI5 Tomatoes - fruit or vegetable?

Are tomatoes a fruit or vegetable? Why is ketchup considered a legal vegetable in California? Why are sun-dried tomatoes safe to eat, while eating a tomato you left on the windowsill for too long would probably make you ill? How can restaurants leave ketchup and mustard out all day but the bottles you buy in the store say to refrigerate after opening?

... we explain like I'm five

Thank you to the r/explainlikeimfive community and in particular the following users whose questions and comments formed the basis of this discussion: spczippo, cbd03b, lolgobbz, kgisaboss, amoeba, sadoni, oneandonlyjackschitt and barrysquatter

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