The difference between absurd reasoning and contraposal reasoning
I've never heard those terms in the context of computer science.  When you mention "contraposal reasoning", I've never heard that term before, but I suspect that might be a reference to proving the contrapositive.  When you mention "absurd reasoning", I've never heard anyone use that specific phrase, but I suspect you might be referring to proof by contradiction, also known as reductio ab absurdum, which is Latin for reduction to absurdity.

These are general proof techniques. They're not specific to showing that a language is not regular.



General advice: I recommend that you try to avoid proof by contradiction.  It's a valid proof technique, but my experience is that it often leads beginners awry and causes them to make mistakes in the proof.  I recommend instead proving the contrapositive; it leads to more straightforward proofs where it is easier to avoid errors, and spot where you went wrong if you do make an error.