Yes, you're 100% right.
A good collector friend of mine in England has a variety of them but believes that the French make is for French North African troops.
The Ottoman buttons are mostly backmarked AU Wien, while the later Egyptian Army (British Mandate) are by Mahfouz Freres Alexandrie, F. Phillips & Co. Cairo or Firmin London.
There was a recent one on eBay which had a star and crescent with rope/cable edge.
Now that's another head-scrather. But I gave it a pass as it went above my budget.
But these are plain sailing compared to trying to identify buttons of past and even recent Arabian/Middle Eastern countries and their armed/police forces. Even Arabic speaking persons could not identify some of the script!
Anyway, it was nice sharing ideas with you. You can always leave me a message on this blog if needed or seach for my name on the net.