What is shown in the cases you mention is not that the model is a model of ZFC, made as a single statement, but rather the scheme of statements that the model satisfies every individual axiom of ZFC, as a separate statement for each axiom.
The difference is between asserting "$L$ is a model of ZFC" and the scheme of statements "$L$ satisfies $\phi$" for every axiom $\phi$ of ZFC.
This difference means that from the scheme, you cannot deduce Con(ZFC).
For the proof that Con(ZF) implies Con(ZFC), one assumes Con(ZF), and so there is a set model $M$ of ZF. The $L$ of this model, which is a class in $M$ but a set for us in the meta-theory, is a model of ZFC, since it satisfies every individual axiom of ZFC. So we've got a model of ZFC, and thus Con(ZFC).
