Is it correct that the P3 Factory use a C1-Factory instead of Y2-Factory? And the module slot count of of them, 3, 3, and 2...

it's not perfect but yes. Y2 has fluid and 3 slots and cost only 1 Science. P3 has no fluid and 2 slots and cost 3 Science.
so both ways are a upgrade in different stages. you need to think about using 3 or 2 production-modules or 3/2 speed modules. 3 production can be more useful as 2 ... but sometimes you want only speed.
but i will think about this again.

Got it, So it isn't just tier 1, 2 and 3, is more like tier 1 than upgrade choice for some situation. It is not bad, I was jut thinking that was a type mistake or copy past hahha.