What is the difference between a contact point and a boundary point on a metric space ?
In topology and mathematics in general, the boundary of a subset S of a topological space X is the set of points which can be approached both from S and from the outside of S. More precisely, it is the set of points in the closure of S not belonging to the interior of S. An element of the boundary of S is called a boundary point of S.
In mathematics, an adherent point (also closure point or point of closure or contact point)[1] of a subset A of a topological space X, is a point x in X such that every open set containing x contains at least one point of A. A point x is an adherent point for A if and only if x is in the closure of A.
Well following your definitions you have that contact point equals closure point and boundary equals closure minus interior. So the difference between closure and boundary is as that the former includes the interior points.
For example the boundary of the interval $(0,1)$ is $\{0,1\}$ but the closure is $[0,1]$.
For a definable topological space $X$ and a subset $A\subseteq X$ let $cl(A)$ denote the closure of $A$ and $bd(A)$ its boundary. Then $bd(A)=cl(A)\cap cl(X\setminus A)$.