This is a simple topology fact that I have forgotten. See Hatcher early in section 1.3., Lifting Properties.
Lemma
Let \(p:\widetilde X\to X\) be a covering space. Then for each path \(f:I\to X\) and each lift \(\widetilde x_0\) of the starting point \(f(0)\), there is a unique path \(\widetilde f:I\to \widetilde X\) lifting \(f\) starting at \(\widetilde x\). In particular, if \(\widetilde f\) and \(\widetilde f'\) are two lifts of a path \(f:I\to X\), then they are homotopic.
Proof
Take \(Y = f(0) = \operatorname{pt}\) in the homotopy lifting property, then the result is immediate.