3-6 Cage (corrected)
Tonight, as I was pulling apart my 3-6 cage and attempting to make it into a Hamiltonian cubic graph, I realized it was in fact not a 3-6 cage. A cycle of length 5 appeared that had not been obvious to me before (highlighted below).
After I had constructed what I thought was a 6-3 Cage, I looked it up online to compare the common construction to mine. Because of that, I already know how to construct the 6-3 Cage, so I included it below.