Graphs may be the wrong structure to use for the problem you are trying to solve.
Graphs only contain information about edges and connectivity.
The three graphs are the same by definition. A graph is planar if there exists a planar embedding.
Perhaps you are asking whether you can find an embedding where the boundary is given by a specific cycle?
I cannot think of a trivial way to change the boyer-myrvold algorithm for this case because there is no concept of "boundary" in the algorithm. It just tries to find K5 or K3,3.
One way that will work, but may not always be possible is this. Choose your boundary cycle. Find the planar embedding by the algorithm. If you see a path that goes out of the boundary, artificially add a new path that prevents the path from taking that route. Then perform the algorithm again.
e.g. your external cycle is abcda, and the "bad" path goes from a to c, the add a new path bvd (where v is a new vertex).
Note, for graph embeddings, internal and external can always be inverted. There is no way around that in any algorithm. However, you can easily transform it by mapping the embedding on a sphere (then internal and external are symmetric).