does epsilon-delta definition of limit presuppose that the function is defined everywhere at (x-delta,x+delta)? definition of limit says that we can choose a delta etc... f(x)-Limit is smaller than epsilon. but the notation f(x) presuposses that f is defined at x.
so does definition say we can find delta where the function defined everywhere at (x-delta, x+delta) such that f(x)-Limit is smaller than epsilon  or we can find delta if f(x) defined then f(x)-Limit is smaller than epsilon
what is the difference? according to one we also must show that the chosen delta deliniates a interval where f(x) is defined
and according to other we should only care about the cases where it is defined.
 A: No, it doesn't: you have the property in the intersection of the interval of radius $\delta$ around $x$ with the domain of the function. This might not be the entire interval. For example, a function defined on $[0,1]$ can be continuous at $0$, in which case the relevant interval is $[0,\delta)$.
A: Does epsilon-delta definition of limit presuppose that the function is defined everywhere at $(x-\delta,x+\delta)$?
Not. The definition only presuppose that, for each $\delta>0$, the function is defined at least in a point of the interval $(x-\delta,x+\delta)$.
More precisely, in the context of real functions of real variables, the definition of
$$\lim_{x\to c}f(x)=L$$
presuppose that, for every $\delta>0$ there exists $x\in \operatorname{Dom}(f)\cap(c-\delta ,c+\delta)$. In other words, this definition presuppose
$$\operatorname{Dom}(f)\cap(c-\delta ,c+\delta)\neq \varnothing,\quad\forall\ \delta>0.$$
It means that $c$ has to be a limit point of $\operatorname{Dom}(f)$. In general, in the context of Calculus, this condition is automatically satisfied so that you don't have to worry about it.
So does definition say we can find delta where the function defined everywhere at (x-delta, x+delta) such that f(x)-Limit is smaller than epsilon or we can find delta if f(x) defined then f(x)-Limit is smaller than epsilon
Your second possibility is correct: you have to find a $\delta$ such that: if $f(x)$ is defined (that is, if $x\in \operatorname{Dom}(f)\cap(c-\delta,c+\delta)$), then f(x)-Limit is smaller than epsilon. You don't have to care about the points where the function is undefined.
