Curves in a plane region

Let X be the subspace of the plane R² shown in black in the pictures below. The pictures illustrate some homotopies between loops in X. Recall that a loop is a continuous function u:[0,1] -> X such that u(0)=u(1). The points on the loop are colour-coded: the reddest point is u(0)=u(1), the blue-green point is u(0.5), the yellow point is u(0.2), and so on.
The first picture shows a homotopy between two curves.

The second picture shows another homotopy between two loops, say u and v. Here the images of u and v are the same set, but the colourings are different, which means that u and v are different functions and so count as different loops.

The third picture is a reminder that he map u:[0,1] -> X need not be injective, or in other words that a loop is allowed to cross over itself.

The final picture shows two loops (say u and v) and a homotopy between them, where the intermediate stages are non-closed curves. Thus u and v are homotopic as curves, but they are not homotopic as loops.