There are two files in my library: the 2.2.0 installer and the 1.1.3 patch. I think I installed the game with the 2.0.0.2 installer because that's the one I have downloaded. Do I need to download the new installer plus the patch to update or will just the patch do?

I have only ever installed the first installer and updated with the three patches.

The patches seem to exchange the very same files, so it's pretty irrelevant which version of the game you have installed or how many patches you have applied before.

Just download the 1.1.3 patch and apply it to whatever version of Chapters you have installed on your system. It should work.