Multiple of Hamiltonian Cycles
Recall a Hamiltonian cycle visits each vertex of a graph exactly once. Thus, the cycles H1 and H2 in your second example can't be Hamiltonian cycles in the same graph.

In your first example, the two cycles could be distinct Hamiltonian cycles in e.g. the complete graph on 4 vertices. That is to say, there can definitely be multiple Hamiltonian cycles in a graph.

If you care about naming conventions, you can call two cycles that don't share any vertices vertex-disjoint cycles. Similarly, two cycles can be edge-disjoint if they don't share edges.