Tach does it again. I'm sitting here scratching my head, dissecting your circle to figure out how you did that! I'm thinking the easiest way is to intersect a plane with a sphere and use "Difference" to cut the hole. The other one looks like adding a circle with 8 vertices, extrude it, fill the center with quads, then add a second circle on the outside with 24 vertices and join the two meshes. Am I even remotely close?
BTW, this has nothing to do with UV mapping. which is what the question is about. We still have to map to the outside vertices. The inside is irrelevant. Where Tach's circle can make a huge difference is in vertex painting or in more advanced UV mapping where each face is individually mapped.
Excuse me while I go off to learn how to make a circle all over again

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