I'm not sure if my question make sense, but it would also be interesting to know if it didn't, so I will ask anyway:

There exist a countable model of ZF. This means (if I understand it correctly) that we can find a model of ZF in ZF, such that the set of elements in the model is countable seen from the outer ZF.

My question is: Can every model of ZF be "seen from the outside" in a way that makes it countable?

It seems to me, that if we have a model A of ZF, the model will be a element in a countable model B of ZF. Now, if you look at B from "the outside" A is countable.