Difference between k-coloring and k-colorable?

Solution 1:

(writing for future readers)

You are right. If a graph has a $k$-coloring then it is also a $(k+1)$-coloring (which does not use colour $k+1$).
(There is a minor technical difficulty here regarding range of the coloring if we define colouring as a function; but this is not important)

A graph is said to be $k$-colourable if it has a colouring.

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