Tri98 Rem and Vash's attempts at getting the gun from Rowan and Zazie relies on them appealing to a person's better nature, like this isn't you/you don't want to do this/no one has the right to kill another etc., which feels more self serving for their own ideologies rather than an appeal to the person standing in front of them.
When Meryl reaches for the gun from the man that wants to kill Vash she appeals to who the man was, ie you were not born wanting to hurt people, and when she is countered with the man's genuine pain she notes that if we don't stop the cycle of violence now it will continue, validating his pain, empathizing with him, and rationalizing that there will be endless suffering if they didn't stop hurting each other now. She doesn't project a better version of this man onto him or try to appeal to that version. She takes the reality of the man in front of her and appeals to who he was to show him who he could be (or return to) by putting the gun down.
But beyond that, she's rationalizing Vash's belief system through her own experiences and not the other way around. That's why it works out for her and why Vash has his breakthrough at the end of the episode/knows how to handle Knives using his own sense of empathy.
(Though that does call into question Vash's previous actions as not being derived from empathy as he views/saves the lives of others to reaffirm Rem's Eden which is scary but really interesting...)