"The first order" IS the imperials tho, just saying it's different is silly. Anyway calling it "the star killer" in the same vein as "the sun crusher" or "galaxy gun" would have been better. But they didn't because that's a silly, nonutilitarian name. Thus breaking the fourth wall and relegating it to "just a reference"
also personally I hate Starkiller he is too OP and too important (like the whole Rebel alliance started because of him???) it didn't make any sense but that's a separate thing.