I understand how we can shrink a circle to a point. Just lower the radius, and the limit to radius $0$ I suppose.
However, how can we deform a point into a loop? The deformation theorem says that two loop contours can be deformed between another if each intermediate contour is a loop. How is a point a loop? And how can I map a singular point to a whole circle? Isn't that not a function?
What I (maybe?) realize is that let's say we have a point $z=0$, which we'd like to deform to $r=1$.
We can write this $z=0$ as the loop given by $\displaystyle z(t)=0e^{2\pi it}$ for $0<t<1$. Then, the deformation function could be $z(s,t)=se^{2\pi it}$. While it makes more sense using this notion, a point doesn't look like a loop because it is a point. Would it be better to think of a point as the limit of a shrinking loop?
A point $p$ determines the constant loop, namely which maps every $t\in [0,1]$ to $p$.
Note that this coincides with the formula of circles when we take the limit $r\to 0$, as you wrote, the corresponding loop for $p=0\in\Bbb C$ is given by $$f(t) =0\cdot e^{2\pi ti}=0$$