Why did John make such a big deal about being a faster runner than Peter?

The issue that John is addressing is that no one went into the tomb unaccompanied. John arrived first, but did not go in. His testimony is that he did not disturb anything before Peter got there.

"...who was behind him" does not emphasize that John was first but that Peter, and not another, was second.

The final reference simply clarified which "other disciple" is being referred to.He is just saying that he one who arrived first followed Peter in, and not another disciple that perhaps arrived third.