Curves in R3\S1

The pictures below show some loops in the space X obtained from R3 by removing a thickened circle. 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 can be regarded as a homotopy in R3 between two loops that lie in X. However, the coloured circle moves through the brown region that does not count as part of X, so the picture cannot be regarded as a homotopy in X.