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.