Can I bring a small ice cooler in Disneyland?

No. You can wrap your pre-frozen water/drinks in aluminum foil to keep them cool and store it in your backpack.

We just brought a bag with water and snacks.

I believe you can we did not but we did bring water bottles in our back pack no problem. I think it is ok as long as it doesn't have wheels

We brought a collapsible cooler with wheels and they allowed it with the exception that we keep it in a locker or tuck it under the baby stroller, only because of the wheels.

I don't those are allowed

Just freeze water bottles and have a good back pack.

No