I have some problems with homotopies.

The situation is this:

Let $X$ be a surface, which is homeomorphic to a 2-Sphere with a finite number (at least 3) of points removed (equivalently, an open Annulus with a finitely many punctures).

$f: X \rightarrow X$ is a homeomorphism, which is isotopic to the identity.

Now,

let's say we have two isotopies $\tilde{f},\tilde{g}: [0,1]\times X \rightarrow X$ of $f$ to the identity, with $$ \tilde{f}(0,x) = \tilde{g}(0,x) = x \quad \forall x \in X,\\ \tilde{f}(1,x) = \tilde{g}(1,x) = f(x) \quad \forall x \in X. $$

Let $x$ be a point in $X$. We can now look at two paths $\alpha, \beta : [0,1] \rightarrow X$, definded by $$ \alpha(t) = \tilde{f}(t,x),\\ \beta(t) = \tilde{g}(t,x). $$

The two paths have the same start- and endpoints, namely $x$ and $f(x)$. It seems to be a well-known fact (I read about it in "Farb, Margalit - A primer on mapping class groups (p.43)"), that on this surface, the maps $\tilde{f}$ and $\tilde{g}$ are homotopic. Thus, there exists a continuous map $h:[0,1]\times [0,1] \times X \rightarrow X$, with $$ h(0,t,x) = \tilde{f}(t,x),\\ h(1,t,x) = \tilde{g}(t,x), \quad \forall t \in [0,1], \forall x \in X. $$

It is clear now, that $h$ gives us a free homotopy of the two paths $\alpha$ and $\beta$, by sending $(s,t) \mapsto h(s,t,x)$ but I don't see a reason, why $\alpha$ and $\beta$ should be homotopic in the usual sense of a homotopy between paths (i.e. fixing the endpoints). My problem is, that this is exactly what is claimed to be true in a paper by John Franks (page 4).

Also, on the open annulus $A = (0,1) \times \mathbb{R}/ \mathbb{Z}$, this is clearly not true: Let $f$ be the identity on $A$. Then, there are two isotopies $\tilde{f}$ and $\tilde{g}$ of $f$ to itself: $$ \tilde{f}(t,[x]) = [x + t],\\ \tilde{g}(t,[x]) = [x], $$ where $[x]$ is an equivalence class in $\mathbb{R}/ \mathbb{Z}$. Now, $\alpha$ is a loop that winds around the hole in the center and $\beta$ is the constant loop. These two are clearly not homotopic in the sense of a path-homotopy, so the argument has to stem somehow from the number of punctures... I'm confused. Any help is much appreciated!