Stihl have come back to me about the chain you need and they have told me this is the correct chain for your machine and bar:
This chain is correct for the MS181 as long as you have a 14" guide bar fitted.
This chain should be suitable for you: 3636 003 0044.
Yes these are both the same part.
I have just checked and this chain has a 1.3 gauge so it should be okay for your bar.
The below chain should be suitable;
3670 000 0072 - 14" chainsaw chain
This is the correct part for your chainsaw, part no. 3636 003 0050.
As long as the gauge, pitch and links are the same as for your original then yes is the correct part for your machine.
The chain you need is part no. 577 61 51 22.
I can confirm this part is suitable for you.
050 refers to the pitch of the chain.