I currently have 23's on my bike which are fine but I also have 28's which I might put on. Is there a difference with the PSI and tire width that protects the rims better when hitting small bumps, potholes and the such?

A 28mm tire has roughly* 150% of the volume as a 23mm tire at the same air pressure, so you have more slack for how much air you can lose between rides (if you don't check tire pressure before every ride) and not increase your risk for either a pinch flat or actually denting the rim if you hit something hard and fast. The skinnier the tire the more it pays to be OCD about tire pressure (depending on how shady your riding conditions are, if everything is clean and flat then you have yet more slack around tire pressure)


* very roughly w/ simplifying assumptions for calculating volume of a torus...

I run 28s in the mid 90s and 23s at ~120. In my experience, running 23s protects rims better because I top them off before every ride while I might wait 4+ days before topping off 28s.

Yes, and they ride nicer too. I'd dump those 23s for 28s ASAP.