Learn English – ___, ___, and I am/are…

Cinderella: "Hooray! We're going to the ball!"

Ugly Sister: "No, we're going to the ball. You're staying home to do the housework."

There's no single generic way to contextualise inclusive/exclusive we/us/our - in the above example, the spoken emphasis makes it clear that although Cinderella uses "we" inclusively, the Ugly Sister is definitely using it exclusively.

In other cases you might need to rely on background knowledge of what makes sense (Cameron says "We want to reduce tax rates"), or it might be implicit through other aspects of phrasing ("We will fight you all the way").

Addressing OP's specific question, the (somewhat more extensive) technical term is clusivity, but we also often refer to the inclusive (or exclusive) we.

I'll also just note that in many spoken dialects, inclusive you can include the speaker - "I like this place! They give you whatever you want!".

Obtain cannot take a that clause, so we obtain that is ungrammatical.

You second example is common in technical papers, and treats the equation as a sort of quotation. What you are saying is something like

We obtain "a squared plus b squared equals c squared"

where the grammar inside the quoted string is irrelevant to the main sentence. (Note that the subscript 2 is read squared, not square).

You could achieve a similar effect by saying we obtain the result a squared... or _we obtain the equation a squared ... _.