Sorry, I've just spotted your question 3 days late. The Moser-Erdős question is about how large the density of a unit distance avoiding set can be. The Hadwiger-Nelson question is about how many unit distance avoiding sets are necessary to cover the whole plane.
As David Eppstein phrased it (https://mathstodon.xyz/@11011110/110810118775049296), ours is the independent set version of Hadwiger-Nelson. That is, the relationship between the two questions is analogous to the relationship between graph chromatic number and graph independence number.
If you define independence ratio as the ratio of the independence number and the vertex number, then this is a bit more than just an analogy. If you build a huge grid of very tiny squares, and connect two tiny squares with an edge if there's an unit distance between them, then in the limit, the independence ratio of this graph is the density that we've looked at, and the chromatic number of this graph is the measurable chromatic number of the plane.
