paint(v) means that you should choose a vertex (v) and color all vertices (u) which have the same color as (v) and at the same time there is no vertex that has another color in the simple path between (v) and (u).

Your solution which is counting unconnected black and white components and output the minimum of them should be true.

The editorial is just explaining how you can count those unconnected black and white components in an efficent way (compressing the tree and finding the diameter).