In the book of Mathematical Anaylsis II by Zorich, at page 122, it is given that
Definition: A set E is Jordan-measurable if it is bounded and its boundary has Jordan measure zero.
As Remark 2 shows, the class of Jordan-measurable subsets is precisely the class of admissible sets introduced in Definition 1.
However, in order for Jordan measure to be defined on a set $E$, we only need its boundary to be Lebesgue measure zero, not Jordan measure zero, so I'm little confused about the definition.
To clarify, if a set is admissible, by definition of Jordan measure, we can define the Jordan measure of that set. Similarly, if $E$ is bounded and its boundary is Lebesgue measure zero, by definition, $E$ is admissible, so I do not understand behind the motivation for using Jordan measure zero instead of Lebesgue measure zero in the definition of Jordan measurable sets.