I'd like to define three things and ask if my understanding is correct / if there is anything I am wrong about or missing.
A removable discontinuity exists at $x=a$ when $\lim_{x \to a-} f(x) = \lim_{x \to a+} f(x) = L$ for real $L$, but $f(a) \neq L$. And $f(a)$ may or may not be defined.
A jump discontinuity exists at $x=a$ when $\lim_{x \to a-} f(x) = L_1$ and $\lim_{x \to a+} f(x) = L_2$ where $L_1, L_2$ are both real and $L_1 \neq L_2$. And $f(a)$ may or may not be defined.
An essential/infinite discontinuity exists at $x=a$ when both the left and right limit exist at $a$ and at least one of them tends to $\pm \infty$.
I would change 3 to say "when at least one of them is $\pm\infty$, or doesn't exist". For instance, $f(x)=\sin\frac1x$ has an essential discontinuity at $x=0$.