What is the flaw in my thinking for the graph of this function?
Solution 1:
Your sketch doesn't even look like the graph of a smooth function - remember that it should just be a deformed plane with nowhere vertical tangents. Here's an animation I whipped up that might help your intuition for this example:
In general it's helpful to remember how changes in the topology of the level curves correspond to local features of the function: a loop appearing/disappearing corresponds to a local extremum, while a transition across a self-intersection like you see in this example at $c=0$ corresponds to a saddle point.