Hamiltonian cycle vs co-NP
The complementary version given by Wikipedia is correct.

coNP consists of the complements of the problems in NP. In this case, the universe of discourse (i.e., the encoded instances) is the set of (properly encoded) graphs. The Hamiltonian cycle decision problem contains those graphs which are Hamiltonian; its complement is exactly the set of graphs which are not Hamiltonian. 

You are correct; the two problems do look almost the same. This is a vital realization concerning coNP (or any co-class in fact). The key difference is that, for some problem classes, being able to efficiently check whether a solution is correct (NP) does not necessarily mean you can also efficiently verify all answers are incorrect (coNP) and vice-versa.