In general, how in-depth do graduate schools analyze the difficulty of the classes you take?
When it comes to linear algebra, there are two basic types of course, proof-based and more cookbook, that just goes through the procedures, manipulating matrices, finding eigenvalues, and so on.  (As you described in your question.)  I can't say whether a graduate admissions committee would care which you took, because:


It depends on what field you are applying for.  For example, if you are pre-med, no one will care if you can write a proof.
It depends on the approach of the university you are applying to.


However, I can answer a similar, related question:

How do colleges and universities evaluate the two types of linear algebra classes when making transfer credit decisions?

In this case, there is a clear distinction between the two.  If you took your linear algebra at College A, the institution you are transferring into, call it College B, will look at A's course description carefully.  If the course you took at A was primarily procedural, you won't get credit for B's proof-based course.  And if they don't have a procedural version, it will just come in as an unspecified elective.

Please feel free to specify your field and a particular university you wish to apply to.  No guarantees, but that might help you get a more useful answer.

Editorial notes (feel free to ignore): if you're interested in doing research in STEM, unless the proof-based course is very badly taught, you'd get much more out of taking that one -- even if you don't end up with an A.  Proofs are like green leafy vegetables -- they're good for you!  On the other hand, there are no prizes given out for completing a Bachelor's degree in record time.  Linear algebra with proofs is an important foundational course and deserves to be savored.