Without getting too bogged down into PbtA frameworks, a successful "check" on a
Persuade (vs. PC) in Stonetop, broadly speaking, gives the target PC the choice to either (1) agree to do what the rolling PC wants
or (2) reveal what could persuade them, through whatever method of their choice.
There is a bit more to it. To "press or entice a PC
and they resist" is a condition that must exist in the fiction for the move/check to be triggered. But the check is also dropped if the player of the target PC believes there is nothing that the other PC could do to persuade their character.
One key difference is that Stonetop distinguishes between two types of Persuade: i.e.,
(vs. NPCs) and
(vs. PCs). NPCs would never roll Persuade against PCs in either Stonetop or PbtA games. But two reasons for that is primarily (a) the GM never rolls in PbtA games but also that (b) a failed Presuade (vs. NPCs) roll could involve the NPC turning the tables on the PC.
As to whether that would be Charisma or Wisdom in D&D 5e, I think that the former may reflect sweet talking it out of the target while the latter would reflect discerning it. Stonetop simply uses Charisma to cover the entire range of possibilities.