Homer does use the word Ilium. Il. 5.204-205:

ὣς λίπον, αὐτὰρ πεζὸς ἐς Ἴλιον εἰλήλουθα

τόξοισιν πίσυνος: τὰ δέ μ᾽ οὐκ ἄρ᾽ ἔμελλον ὀνήσειν.

So I left them, and came to Ilion on foot,

Trusting my bow, for all the good it has done.

Lombardo's translation, emphasis mine.

There are other places as well; this is just the one I found on a quick search. The LSJ actually gives line 210 as its example.

In Latin, the hinterland of the city is called "Troas" - or in English, "the Troad." That's where "the Trojan War" comes from. It's like saying in WWII, Germany fought America, not Washington DC.