Base Point Homotopy vs Free Homotopy Example

Does there exists a space X, along with two loops $f,g$ based at $x_0 \in X$ such that $f$ and $g$ are freely homotopic and not "based point" homotopic?

Two loops are freely homotopic if there exists a homotopy between.

A base point preserving homotopy is one where the homotopy between $f$ and $g$ also satisfies $F(x_0,t) = x_0$

Yes there is. In general, the set of homotopy classes of maps $S^1 \to X$ corresponds to the conjugacy classes of $\pi_1(X, x_0)$. So any two elements of $\pi_1(X, x_0)$ which are conjugate but not equal will provide such an example; note, for such elements to exist, $\pi_1(X, x_0)$ must be non-abelian.

It's not a great picture, but the blue and red loops below are freely homotopic, but not homotopic relative to the basepoint $x_0$. In $\pi_1(X, x_0)$, they are conjugate via the green loop.