Yes it’s still a quad as long as it has 4 vertices, even if it looks like a triangle. Be careful doing this on the edges of things tho.
As above the pinching is simply down to too many vertices and the corresponding pole that’s formed. You only need the centre and outer edges on the corner in the last screenshot. You’d be left with triangles but on the flat parts and can probably get away with them there if you don’t want or don’t know how to resolve them further. The other thing to experiment with is using weighted bevels and bevel modifier as part of your Sub-D model workflow.
