Definition $f(x)$ diverges to negative infinity for $x \to a$

Can someone tell me, if the following is a valid definition of $f(x)$ diverging to $-\infty$ for $x\to a$

$f(x)$ diverges to $-\infty$ for $x\to a$, if and only if for all $\epsilon\in\mathbb{R}$, an $x<a$ exists with the property $f(x)<\epsilon$.

No: Where in your definition do you incorporate $a$?

It should rather be the following: $\lim_{x\to a}f(x)=-\infty$ if for each $M\in\mathbb R$, there exists some $\delta>0$ such that $0<|x-a|<\delta$ implies $f(x)<M$.

No, it is not. Note that the number $a$ is not mentioned at all after the “if and only if” part of the definition. So, with this definition, the assertion “$f$ diverges to $-\infty$ as $x\to a$” would not depend upon the choice of $a$.