... that crosses exactly once (only once) each interior line segment of the rectangle, whilst staying inside the rectangle for these 2 rectangles.
My immediate reasoning was to remove all the outside edges, but keep the vertices. As a result, I said no to both (as there was no Euler cycle in the graphs that remained after doing so). Is this a matter of simply removing edges & checking for an Euler cycle as I initially thought? Or should I remove any line segment connecting to the outside perimeter as well?
When a path passes through a vertex, it walks on two of its edges and the only way to pass also on the third one would be if that edge is the last one of the entire path. But since you are looking for a loop, this is impossible. Also relaxing the requirement from "loop" to "path", it is also impossible since there are two odd neighbours vertices.