Question about Strict Preference Relation
You are correct that the motivation regards issues surrounding completeness. As you point out, when $\succcurlyeq$ is complete $y \not\succcurlyeq x$ implies $x \succcurlyeq x$ (hence, $x \succ y$).
If, however, we do not want to assume complete preferences then (2) does not imply (1), but strict preference can still constructed from weak using the more complex definition. (One could also take the set difference of weak preference and indifference).