I think you are missing something. If you are beating against a
southerly wind, then you're heading either SE or SW. If you then turn
180, you'll be on a broad reach heading either NE or NW.

True.

But the plain fact is that there are a lot of things
on boats that have no counterpart in everyday life. If you want to talk
about them, then you need to know the right word(s).

If my wife turned to me in the boat and said "harden up" I'm afraid my only
reply would have to be: "Right here? Right now?" So, why don't you tell me what
it means? I have a feeling it has something to do with sheeting in the main or
jib to keep them from luffing, but I honestly don't know, and as you can see,
it means so much to me.