Generalization of (vertex) coloring of a graph to hypergraphs I'm wondering why is the generalization of a coloring of a graph is:

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*(The correct one) Assigning colors to vertices of a hypergraph so that no edge with cardinality larger or equal to 2 is monochromatic

*(Why not this one) Assigning colors so that each vertex that is connected by an edge have different color

The "2 vertices ANNE and BILL has an edge between them, so they have some relation (that we assigned), and thus they must have different colors (represents something else)" analogue doesn't apply to the definition of coloring a hypergraph, what can I substitute it with?
"These $k$ people (vertices) are related (thus in the same hyperedge), so we must assign at least 2 different things (colors) for them" isn't a very good analogy.
 A: The second generalization (all the vertices on a hyperedge must have different colors) is just a graph coloring problem in disguise. Define a graph by putting an (ordinary, $2$-vertex) edge between any two vertices that are on the same hyperedge. Then we are just trying to find a proper coloring of this graph.
The first generalization, where we only require that the vertices on a hyperedge are not all the same color, is a genuinely new problem. Even if it's a bit harder to think of applications for it, this is the only worthwhile definition, because it's the definition of a problem we couldn't talk about before.
(I'm exaggerating a bit; the "rainbow chromatic number" of a hypergraph, where every vertex on a hyperedge has a different color, does see some use. That's because, even though it reduces to a graph coloring problem, it reduces to a graph coloring problem for a special kind of graph that's a union of cliques, and we might still be able to say something about it that doesn't hold in general.)
