You unlikely get rid of those artifacts on such a low tesselated sphere. The faces are trapezoids which - as you've seen yourself in the 2D case - don't map niceley to rectangular tex-coords.

You could UV unwrap your sphere like in the following picture, but then you still have the problem of painting said texture.



The 2D problem turns up regularly here and elsewhere. There are two solutions: One is a bilinear inverse, the other is the perspective approach. The former is more involved, for the latter one can (ab)use the graphics hardware.

I once found the solution myself using a bit a convoluted approach here. More elegantly one can use homogenous 2D coords, derived here by (of course) Alvaro. IIRC this is the same approach Blinn showed in his Notation3 book.

I think this is also possible for 3D. Since this is a sphere (all faces are symmetric trapezoids) it will likely simplify the calculation.

Edit: Proof of concept.