> NP-complete problems are those that, if any one of them can be solved in polynomial time, then all NP problems can be solved in polynomial time.

That is (nearly) correct under Cook reductions but not Karp reductions.

I say "nearly" because "if language L can be solved (decided) in polynomial time, then so can any problem in NP" is a vacuously true statement for any L not in P, so by this definition nearly all languages (a measure 1 fraction of them) are also NP-complete despite most of them not being in any particularly interesting complexity class.

If we escape this degeneracy by replacing the antecedent with "given access to an efficient way to decide L", then we've recreated NP hardness under Cook reductions.