If Heavensward is anything to go by, no (HW only had two Primals added with the initial storyline, no more were added in any patch after, and we had technically three Primals with SB already). And as others have mentioned, Golem is an enemy type not a Primal here (just because it was a summon in the past does not mean it will be a Primal, as there have been quite a few past summons... that have shown up in FFXIV as just regular enemies or even just npcs and not Primals - Diabolos, Cait Sith, Ark, Gilgamesh, Chuchilian, Ultima, Yojimbo to name a few).

And Anima will never be a summon as it's a minion you get for completing the anima weapon questline (not to mention a term used to describe the body's tolerance for teleporting), so that's out too I'm afraid.