For this to be true, G itself must be planar, and additionally it must be possible to add edges to G, preserving planarity, in order to create a cycle in the augmented graph that passes through each vertex exactly once.
That is, for this alternative definition, it should be possible to add both vertices and edges to G to create a Hamiltonian cycle.
However, if a graph can be made Hamiltonian by the addition of vertices and edges it can also be made Hamiltonian by the addition of edges alone, so this extra freedom does not change the definition.
[3] In a subhamiltonian graph, a subhamiltonian cycle is a cyclic sequence of vertices such that adding an edge between each consecutive pair of vertices in the sequence preserves the planarity of the graph.
[8] If the edges of an arbitrary planar graph are subdivided into paths of length two, the resulting subdivided graph is always subhamiltonian.