Er, no. I don't know where all the pseudo-science about "greens",
"browns", "carbon", "nitrogen" and all that came from, but it really
doesn't describe what is going on. Your description of how to use it
is fine - it is the explanation that isn't.

Grass is not a high-nitrogen material, and the reason that grass
clippings make a slimy, smelly mess is because they compact into
a wet, ANAEROBIC lump. Longer, more mature grass does not do that
on a small scale. Sawdust is less compressible, less decomposable
and prevents the smell by maintaining aeration.

Bullshit
Fresh grass clippings here run a carbon to nitrogen ratio of from
about 9:1 up to 20:1
probably because most folks with lawns just pile on fertilizers
without ever doing a soil test.
An efficient fast and hot compost is blended to have a carbon to
nitrogen ratio of 30:1
Sawdust and clippings will still compost hot at twice this (60:1)
because they are finely divided and have much surface area.

This ain't rocket science, I defy anyone to pile damp organic
materials outdoors and have it not rot. What differs is how hot you
want the pile to get and how fast.