I read the question at Gödel's Incompleteness Theorem -- meta-reasoning "loophole"? about Gödel's incompleteness theorem. My question is little about the contents of that other question. Rather, it is about terminology.
How is the thinking system or formal system called, which is outside of the system described by Gödel's theorem? Are there terms for systems when we create formulations in one system about another system? I mean, are there coined words for "inner system" and "outside system" and a generally accepted "outermost system"?
There is no absolutely strict terminology. You could say either "inner logic" or "inner system," and we'd know what you mean. But these terms should better mean the same thing! Similarly, "outer logic" or "meta logic" are interchangeable.
There can't be an outermost logic independent of context. You wouldn't even want two meta logics, as we (more or less) make up the rules in meta logic. So why do it twice and make your math that much more scattered? Better to encapsulate all of the meta voodoo in one place. Of course, I'm no logician, and there may be a use for it. I'd be interested to see any counter arguments.
That being said, a lot of very important and active work takes place on a meta logic (ZFC mumble mumble, quantification over first order formulas mumble mumble) that sits just on top of first order as the inner logic. This is a pretty normal place to study large cardinals or model theory.