Is there an easy way to do this? I want to put my phone in there, and in
the winter the heating will damage the phone - removing the cup holder
and putting the phone lower will work.

I've fiddled with the insertion points several times for a good half hour,
hoping the holder's inserted with a thin section & will pop out if I hit
it just right but no such luck.

And if anyone knows how to find this answer myself next time,
I'd much appreciate links.

 

Hopefully this answer your question.
This is from a Japanese parts catalogue, so yours will be mirror on this for left hand drive.