This is a definition from WikiBooks:
This is another definition from LibreTexts:
The first definition says except possibly at $x=c$ while the second definition says $f(x)$ is defined for all $x\neq a$.
Why do these two definitions define $f(x)$ differently?
I believe you're having a math-speak issue. In the second definition, "$f(x)$ is defined for all $x\neq a$" doesn't mean that $f(a)$ must not be defined. You have to read the sentence in the broad sense (as with many other situations in math, such as the use of the word 'or' as a logical connective). All we're saying is that we require $f(x)$ to be defined at every point other than $a$. At $a$, we make no requirement that the function be defined.
If $f$ is defined at $a$, then great, good for you, (but the value of $f(a)$ makes no impact as to the limit definition).
If $f$ is not defined at $a$, then that's no trouble either.
Said differently, I read the sentence
"$f(x)$ is defined for all $x\neq a$"
as the one-sided implication
"If $x\neq a$ then $f(x)$ is defined",
NOT as the biconditional
"$f(x)$ is defined if and only if $x\neq a$"
The key point is that in the definition of limit we don't care of the limit point, therefore the function can be or not defined at that point, indeed in the definition (e.g. for finite limit):
$$\Big(\lim_{x\rightarrow a} f(x) = L \Big)\iff \Big(\forall \varepsilon >0\, \exists \delta > 0: \forall x\in D\quad 0<\vert x-a\vert <\delta \implies \vert f(x)-L\vert <\varepsilon \Big)$$
the condition $0<\vert x-a\vert <\delta$ implies that we are assuming $x\neq a$.
Refer also to the related:
