I'm looking for examples of nulhomotopic and non-nulhomotopic maps from $S^1$ to somewhere to better understand why nulhomotopic maps can be extended to a disc.
If we imbed $S^1$ into the punctured plane in the natural way, is that map nulhomoptic? On one hand I say no because you can't move through the punctured point, but perhaps you don't need to, you just have the circle disconnect at some point and the homotopy goes around the punctured point. Is that possible?
Your imbedding is not, in fact, null-homotopic for the intuitive reason you state. A map is null-homotopic if it is homotopic to a constant map; the act of "disconnecting the circle" means you no longer have a homotopy.
I think you can use your example: imbedding $S^1$ into the punctured plane is not null-homotopic, and combine that with the fact that the imbedding of $S^1$ into $\mathbb{R}^2$ is null-homotopic to see why something null-homotopic could be extended to a disk. For instance, in the first map, what is inside the imbedded circle? What is inside the imbedded circle on the second map?