Did a search. After reading a few articles, I think I can summarize it like this:

Menora means lamp, and is usually 7 candles. Hannukah menora is a lamp for hannukah and has 9 candles. It is usually called a hannukiah, but that's hard for non-Hebrew speakers to say, so we usually hear menora or hannukah menora.

Edit: forgot to include the sources. I'll try to find them all, but here's the one that, though the shortest, said the most.

Edit: found two more of the sources (1, 2), but can't refind the longest one. Sometimes, it's very annoying not having history saved.

In my experience, it's not really a term you'll come across unless you're Jewish/around a Jewish family at Hanukkah. I have only ever heard/used this term with my family and other Jews because the generic "menorah" is what's known to the population at large - assuming they know anything about Jews to begin with.

lol wtf is this? I have Jewish heritage and never heard of this

My family have always been fairly pedantic about it. Only call it a hanukkiah. It's a square and rectangles thing, hanukkiahs are menorahs but as a menorah it needs to be able to hold nine candles, with one candle preferably on a higher level than the others to be considered a hanukkiah.

Wait until you find out about the little square hat thing 🤯

I knew of it, but I also went to Hebrew High School once a week for like 6 years. Like CCD style, not full time high school, that was secular for me.

I like the word menorah more, so I use it more!