Logical meaning of "And / Or"

Natural languages are very vague about this.

You don't expect people to do both: Go swimming and playing video games after asking them "Do you want to swim or play video games?". So in this case, we are understanding an OR to be exclusive. But in a different context like "Do you want milk or sugar with your coffee?" you are considering the OR to be inclusive.

I don't think "and/or" is a good approach to differentiate an exclusive OR from an inclusive one. I think it's more intuitive to talk about "or" (inclusive) and "either or" (exclusive). I cannot answer this question regarding laws, because it might differ between countries and I am not into laws.

As @ЯegDwight's comment and @meisterluk's answer tell you, speech is one thing, writing another—and context is everything!

As far as legal use goes: I haven't written for lawyers in thirty years, so i'm no longer aware of how much notice courts take of linguistic change.

But other things being equal, and/or is both lazy and graceless. Taking the trouble to say exactly what you do mean and exclude everything you don't mean is what lawyers get paid for, and shortcuts will eventually bite the practitioner in a sensitive part.