In German Mills is known as Mühle - which can be translated as 'grinder' or 'mill' as in the machine that grinds.

In the past, and even today the action of moving the pieces is referred to as grinding - especially the back and forth movement between two Mills.

This is one possibility outside the possible corruption of the Latin words.

I don't have it on hand but the answer is probably in H.R Murray's book the History of board games other than Chess. It has an entire section on the game including how it travelled to various countries.

I'll try to find my copy and have a look.