Let $f:\mathbb{R}^n\to\mathbb{R}^m$ be given such that it is continuously differentiable. According to Wikipedia, a "critical point" of $f$ is a point $p\in\mathbb{R}^n$ such that:
1) According to one paragraph it is the condition that $\partial_j f_i = 0$ for all $j\in\{1,\dots,n\}$ and $i\in\{1,\dots,m\}$.
2) According to another paragraph, it is the condition that the matrix with the $(i,j)$ component $\partial_j f_i$ (i.e. the Jacobian matrix) has rank less than $m$.
It seems like these two definitions are not equivalent. Why are there two different definitions? I can understand why the first one is called "critical point": it gives exactly the points which are local maximum, minimum or saddle points. Why is the second one called "critical point"? What does it give, except for saying that the function at this point is not surjective?