The motivation for defining Brouwer degree as $\deg(F,\Omega, y_0)= \sum_{x\in F^{-1}(y_0)} \operatorname{sign} J_{F(x)}$

Question

Brouwer degree is defined as $$\deg(F,\Omega, y_0)= \sum_{x\in F^{-1}(y_0)} \operatorname{sign} J_{F(x)}$$ This definition can be traced back to the paper of Nagumo 1951. I am wondering why we want to define it in this way, why we can’t instead define $t$ as the number of solutions