DEF 1:A function $f:\mathbb{D}\to\mathbb{C}$ is said to have a nontangential limit at a point $z\in\partial\mathbb{D}$ if, for some $\zeta\in\mathbb{C}$, any time a boundary path $\gamma:[0,1)\to\mathbb{D}$ approaches $z$ and lies entirely in some Stolz angle having $z$ as its vertex, then $f(\gamma(t))\to\zeta$ as $t\to1^-$.
Why is this definition made in terms of paths, and not just something like:
DEF 2:$f(z)$ has nontangential limit $\zeta$ at $z$ if $f$ converges (normal $\epsilon\ \&\ \delta$ definition, no paths involved) to $\zeta$ in every Stolz angle at $z$.
I know there is an example of an unbounded analytic function which satisfies Def 1 but not Def 2 (take an entire function which converges to $0$ along every line, and transport it from the half plane to the disk).
QUESTION Are Def 1 and Def 2 equivalent for bounded analytic functions?
I'm not sure why you have "uniformly" in definition 2. I would just say $\lim f(w) = \zeta$ as $w\to z$ within each fixed Stolz angle with vertex $z.$
The two definitions are equivalent. Definition 2 is the same as saying that for each Stolz angle $\Gamma$ at $z$ and each sequence $z_n$ in $\Gamma$ that tends to $z,$ we have $f(z_n) \to \zeta.$ If definition 1 holds, and we have a sequence $z_n$ as above, just create a path that goes through $z_1,z_2, \dots$ in order to see that 2 holds. If definition $2$ holds, suppose 1 fails. Then there is a sequence along the path on which $f$ fails to have a limit, contradiction.
The entire function that goes to $0$ along each straight line will not lead to a counterexample, because the rates of vanishing vary from line to line.