Doing some research I am pretty sure the first graph is a square antiprism graph but I'm not sure about the second graph. In the second graph every intersection should have a "node" covering it, including the four corner points. To make the first graph into the second graph you have to add a lot more lines that "go across the box and connect opposite corners."
The second graph is a much more built up representation of the first graph. In the first graph every node has degree 4. In the second graph every "interior" node has degree four and each of the 4 outer corners has degree 30, but you can increase the degree of the 4 outer nodes by any amount by adding more edges. So by adding more edges and nodes the degree of the interior nodes stays constant at 4 while the degree of the outer 4 nodes increases. Is there a name for this type of graph and has it been studied in the literature before?
If you have a rectangular set of points (vertices) and connect horizontal or vertical adjacent vertices you get what is called a Grid Graph. In your case, you add one extra vertex for each of the four sides of the rectangle and connect these new vertices to two adjacent vertices to form a four cycle and you also connect each new vertex to all the vertices of the corresponding side of the grid graph. I would call this an augmented grid graph, although at least one brief article has used that name for a grid graph with some extra edges between neighboring vertices but no new vertices.