It is crucial to use the right notion of reduction to define completeness inside NP. Different notions of completeness inside NP may have significant impact on the properties of complete languages. Initially, P vs NP problem was formulated by Cook using polynomial-time Turing reductions. Later Karp refined the notion of completeness using polynomial-time many-one reductions. It is not known whether complete problems are equivalent under different notions of completeness ( Cook vs Karp reductions).
This post on CS Theory shows that defining completeness using injective Karp reductions would prove $P \ne NP$. The reason is that SAT is dense language and can not be reduced to a sparse language using injective Karp reductions. Mahaney's Theorem states that $P=NP$ if and only if there is a Karp reduction from SAT to a sparse language.
Also, one of the answers highlights the fact that all known Karp reductions to NP-complete problems are injective. There is no known natural NP-complete problem via Karp reduction that can not be made one-to-one. If Cook had defined NP-completeness using injective Karp reductions then the P vs NP problem could not existed.
What arguments do exist against defining completeness in NP using injective Karp reductions?
Proving the existence (or non-existence) of polynomial-time algorithms for SAT would be more satisfactory way to resolve P vs NP but it is important to link it to the existence (or non-existence) of incomplete languages inside NP. Ladner's theorem states the $P\ne NP$ if and only if there exists an incomplete language in $NP$ \ $P$ (under Karp reductions).