I would like understand how this task is meant:
"Show that for self-maps of $S^1$ free and based homotopy are the same."
My interpretation would be that if we take arbitrary $f, g: S^1 \rightarrow S^1$ with $f(1) = g(1) = 1 \in S^1$, then free homotopy of $f,g$ would imply based homotopy relatively to $1 \in S^1$. But this can not be true, since $S^1$ is path connected (so that all such maps $f,g$ are homotopic, but not relatively homotopic, since $\Pi(S^1,1) = \mathbb{Z})$.