Does the size of the rain collecting vessel matter? In other words, does its diameter matter?

Nope, any diameter will produce the same result.

Why? the height of the water in the collector is equal to the volume of water in it divided by the area of its opening. The volume of water in the collector is proportional to the rate of rainfall (measured in something like drops per unit area) times the area of the can. If we put these together, we find that the area of the rain gauge cancels from the top and bottom of the ratio—meaning the area doesn’t matter!

The only meaningful measure left is the height of the rain collected. Hence only mm or inches.

Another way to think about it is – if all the rain that fell didn’t disappear (absorption in the ground, evaporation, etc.), and if there were a way to keep it in one location like it is in a vessel then it’s height would stand at x mm, or inches, if you are like me :), in that location.

So, stating the measurement as volume would be misleading unless you also specified the amount of area on which that volume fell.