I can't speak for its usage in the US, but in the UK, it means different things to different people. For me (born and bred in the south-east of England), the three main meals of the day are breakfast, lunch and dinner. For some of my friends, from different parts of the UK, the three main meals are breakfast, dinner and tea. Others use breakfast, lunch and tea. Some use "supper" for their evening meal. I do not. For me, supper is a light snack some people eat a couple of hours after their main evening meal. If someone invited me to "supper", I would find it rather odd. I would assume they were inviting me to their house (we don't eat supper at a restaurant), and that I would have to get there at maybe 9pm or 10pm and just have perhaps some bread and cheese, or a biscuit and a glass of milk!
If a man was asking me out on a date, I would expect to hear "Would you like to go out for dinner with me sometime?"
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