That same thing happens to me sometimes. I wouldnt worry about it. Your armor may just be lagging.
To answer you question, 'Do you have different ranks for each playlist?', yes. Say you play Team Slayer and Multi Team. In Team Slayer say your a Seargant and in Multi Team a recruit. It is not the same thing (of course). Playlists do not have ANYTHING connected within each other in terms of ranks. Say your in Multi Team and you look at your service record and it shows your a Seargant. Thats just your highest rank in the playlists you participated in.
Did I help with both? Sorry if your confused. I tried to put it in the best way possible.