Triangles are polygons. Quads are polygons. Faces with 2841 edges are polygons. 'Polygon' is a bit of a vague word, like saying the word 'coin'. Use triangles if you want to specific, or verts if you really want to be precise.
One quad is two triangles, a five sided polygon is three, and so on. You won't save any polygons because it's all triangles in the end. Sometimes people display their polycount in quads which can be misleading as you said. I find it's better to say "this model is 2000 tris" rather than polygons to avoid confusion.
poly!=tris a polygon can have more than 3 sides, a triangle has only 3, and it is what polygons are cut into so the gpu can render them, since gpus only render tris.
Are you saying that sometimes models that you import are triangulated? If you're importing it from a game it will be triangulated because the renderer can't interpret quads.In most other cases a mesh with mostly quads will be better, as it is easier to work with. When you talk about polycounts in a game you are always…