What is the relationship between two definitions of Turing-computability of a partial function?
Definition 1 is incorrect. It does not define the notion of partial computable function correctly. Definition 2 is correct.

Proposition: A partial function $f$ is computable according to your definition 1 if, and only if, it is a partial computable function whose domain $\mathrm{dom}(f)$ is a computably decidable set.

Proof.
Suppose $f$ is computable according to definition 1. Then for every $n \in \mathbb{N}$, we have $n \in \mathrm{dom}(f)$ if, and only if, $\bar{f}(n) \neq \bot$. So we can decide whether $n \in \mathrm{dom}(f)$.

Conversely, suppose $f$ is a partial computable function whose domain $\mathrm{dom}(f)$ is decidable. Then $\bar{f}$ is computable because we can compute it like this: given input $n$, if $n \in \mathrm{dom}(f)$ then output $\bot$, otherwise output $f(n)$. $\Box$