What does "claiming the lane" mean? If one is "claiming the lane" does that mean: a) they simply ride on the street (i.e. on the left side of the lane in the US, Germany, France, ..) or b) they are riding in the middle of the lane, with someone else side by side, or some other way that they take over most of the lane width? or C) Another definition?

b) is most correct, except that you don't need anyone else by your side.

Taking the lane is riding pretty much right in the middle of the lane, and asserting control of the entire lane. Riding in the middle makes it obvious to motorists that they'll have to change lanes, or wait for a safe opportunity, in order to pass you.