Difference between these two definitions of limits 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 $x\neq a$"

