@a547f8fb wrote: " I wonder how we can distinguish the black and yellow regions in your figure, since they are combined into the fundamental region of SL(2, Z)."

If you read my article you'll see the answer to this explained in detail:

https://johncarlosbaez.wordpress.com/2023/09/23/the-moduli-space-of-acute-triangles/

In brief the answer is GL(2,Z), a group which is 'twice as big' as SL(2,Z).