Why does the texture skew when you subdivide the plane once?
Each quad face is always made of 2 triangles in 3D softwares. I guess the result you see has to do with the way Blender will triangulate each of the 4 faces it creates when it subdivides.
Here I use a grid as image texture to make it more understandable. On the left, the original object, on the right, subdivided once. Here is how Blender triangulates:
If I select my original object, cut edges the same way to make only triangles, and turn the outline into a circle, I'll have the same result:
It would work better that way:
With a higher subdivision, the problem tends to be less noticeable.