Saying it’s a Jedi’s weapon doesn’t mean he’s saying it’s exclusive to the Jedi.

I don’t understand this… we aren’t supposed to believe a “Jedi weapon” is a “Jedi weapon”? So what does it mean then? Was the Emperor supposed to say, “this is a Jedi weapon and no other group of people can use it ever, for any purpose” for the intent of his statement to be clear?

Why would he not just call the lightsaber a lightsaber, or not discuss it at all?

I think the use and discussion of lightsabers throughout the films make it abundantly clear what the intent is.