I read one definition of a thin loop: $\gamma$ is a thin loop if there exists a homotopy of $\gamma$ to the trivial loop with the image of the homotopy lying entirely within the image of $\gamma$.
Because a homotopy is a continuous map, I guess an example of a thin loop might be as following:
$\gamma(t)=\left\{ \begin{array}{cc} \sigma(2t), & t\in[0,1/2)\\ \sigma(2-2t), &t\in[1/2,1] \end{array}\right.$
where $\sigma(t)$ is any continuous curve.
Are there any other examples?
At the same time, two loops $\gamma_1$ and $\gamma_2$ are called thinly equivalent if $\gamma_1\circ \gamma_2^{-1}$ is a thin loop. So by this definition, I expect that if $\gamma_1\circ \gamma_2^{-1}$, $\gamma_1$ and $\gamma_2$ must be thin. This does not sound correct, but I cannot image other possibilities.
To add to my comment above. In general, it is clear that a map $f$ is homotopic to a constant map with homotopy inside its own image if and only if the image of $f$ is contractible. Therefore, for this case a loop $\gamma\colon S^1\rightarrow X$ is a thin loop if any only if its image is contractible. Therefore it must of the form you have already stated (up to some scaling say), otherwise the image of $\gamma$ would be homotopy equivalent to a circle which is not contractible.