From my understanding,
Proof by contrapositive: Prove $P \implies Q$, by proving that $\neg Q \implies \neg P$ since they are logically equivalent.
Proof by contradiction: Prove $P \implies Q$ by showing that $P \wedge \neg Q$ yields an absurdity and hence false. So $\neg (P \wedge \neg Q)$ is equivalent to $\neg (\neg (P \implies Q))$ and $P \implies Q$ by double negation so showing that $\neg (P \wedge \neg Q)$ proves $P \implies Q$.
If the absurdity derived during the procedure for a proof by contradiction is $P \wedge \neg Q \implies\neg P$, we have essentially already proven $P \implies Q$ by contrapositive since $\neg Q \implies \neg P$ is precisely the required condition for proof by contrapositive. But $(P \wedge \neg Q) \implies \neg P$ is also a contradictory statement which means that $P \implies Q$ must be true.
Now the question is this. Is this proof by contradiction still a valid form of proof even though its a proof by contrapositive in disguise? To me, this proof by contradiction also seems to be a valid proof as it does seem to satisfy the conditions(if they are correct) for proof by contradiction.
Additionally, if you have a contrapositive proof, so you have shown that $\neg Q \implies \neg P$, is it possible to rephrase this in a proof by contradiction by supposing that $P \wedge \neg Q$ instead of just $\neg Q$.
If this is the case, what is the point in distinguishing proof by contradiction from proof by contrapositive?
edit: My thought is that proof by contrapositive is a direct proof while proof by contradiction, in this case, depends on the validity of the double negation law which apparently isn't valid in intuitionistic logic.