Not really a Tanach person, but I’m not seeing your question.
Part of the city of Yerushalayim is a mountain called Har Tzion. It had a fortress on it. Dovid conquered that fortress and made the mountain his administrative headquarters (i.e. Ir David). As a result Tzion became shorthand for the entire city.
The Beis HaMikdash was on a different mountain, Har HaMoriah, that also was part of Yerushalyaim. Moriah is also used in Tanach as a shorthand for the city. (e.g. Hashem’s instructions to Avraham by the Akeidah).
Whatever Tzion is, if you must insist it’s not Yerushalayim, it’s certainly not a building that was located on a different mountain.