Sorry for that! :-( I don't know much how the point system works here on MO, but if I can do anything to bump your question up, I'll do. It's a good one!

My bad here. There was a typo in the notation for $d$ in my original question. Then I changed it, but I didn't realize that meanwhile Long had already posted! By the way, thank you @HailongDao for your answer, very interesting that the only non-trivial cases are in dimensions $2$ and $3$

Nice reference! A compact version of the equivalence can be found also in Section 3 of the following paper by Burban and Drozd: arxiv.org/pdf/0803.0117.pdf

I also suspect that the original result should be due to Klein, but couldn't find a precise reference yet. Thank you anyway!

Thank you Nick! Beauville's paper completely answers my question about what happens over arbitrary fields. It is really nice! However, the classification over $\mathbb{C}$ is much older of course. I am still interested to find (for historical reasons) an older reference for that. Do you know one?

@M.Shahryari Thank you! Although I am mainly interested to the field situation, I would be happy to have a look at the division ring case as well. I couldn't find Shahabi's Ph.D. thesis unfortunately.

@GeoffRobinson Thank you! It is indeed a nice argument to reduce to the case $G\subset SL(2,\mathbb{k})$.