I don't believe that system 1 and system 2 are good names. I do understand the motivation of using names that aren't already mapped with meaning but there are other ways.

You can use Greek and Latin syllable to make up new terms.

I recently learned about Piaget's stages of learning: Assimiliation, akkommodation and equilibration. Those words don't really create conflicts with existing concepts. At the same time the names make more than than a numeric ordering.

Philosophers have already named many concepts. Often it's better to use on of the existing names than to make up a new one.