The OpenGL specification basically had no explanation of what a "Quad" was, it didn't say anything about how to render quads, interpolate quads, etc. Pretty much every driver implemented quads as a funky way of spelling "two triangles". Much like tristrips/trifans/etc., index buffers give you all of the benefits with none of the drawbacks.

> For instance, the X11 XRender API explicitly supports drawing trapezoids.

It should be said XRender trapezoids have extra constraints: the top and bottom edges of the trapezoid must be completely straight (go exactly left to right). So it's basically a rectangle with sloped left/right sides. This makes it extremely easily to software rasterize, while you can still break each trapezoid into two triangles for 3D hardware.