Since I've seen pictures of both kinds, was one color used in certain circumstances and one in other situations, or was the use of either kind based on whatever happened to be available?

Hi
The White MP helmets were usually worn a rear area's eg: Main Base camps & the likes of London ( there's some pic's of MP's in London near the Albert Hall ).
The OD helmet with the white band & white MP letters was worn in combat area's.

Ah, that makes the most sense.