I'm reading the total derivative as a linear map definition on Wikipedia. The function in the definition is $f:U \mapsto \mathbb{R}^m$, where $U$ is an open set in $\mathbb{R}^n$. However, the derivative is defined as a linear map that maps from $\mathbb{R}^n$ to $\mathbb{R}^m$.
Does that mean the linear map has a different domain (i.e. $\mathbb{R}^n$) than the function itself (the function has domain $U$)? How is that possible? In my understanding, the total derivative, when given a point from the domain of the function itself (i.e. $U$), tells us the speed and direction of the function at that point. If so, it doesn't make sense to put a point that's in $\mathbb{R}^n$ but not in $U$ into the total derivative, does it?
No, the derivative is defined just in the domain of the function. In other words: how you could define the derivative of $f$ where there is no $f$?
You are mixing the concept of derivative of a function with the concept of derivative of a function at a point. Observe that if $f:U\to\Bbb R^m$ for some open $U\subset\Bbb R^n$ is differentiable then
$$\partial f: U\to\mathcal L(\Bbb R^n,\Bbb R^m)$$
and $\partial f(x)\in\mathcal L(\Bbb R^n,\Bbb R^m)$ for some $x\in U$. You see the difference?
However you can expand the domain of the function that defines a derivative, if you wish, whenever the expansion defines a function properly. But this new function is not anymore the derivative of the original function.