In universe it makes little sense for her to go by either of them at the moment, though I think X-23 should be kept for branding purposes.
I think a lot of people see her being wolverine as some kind of promotion, when it's really not; Logan was and is always going to be "the Wolverine" in peoples eyes, unless maybe he'd stayed dead for 20 years while she kept the title (though look how well that's working out for Wally West.); if she keeps the name she just becomes "that other one", no identity or brand of her own, forever lost in Logan's shadow; not to mention it just being confusing on a basic level on who you're talking about. (I will admit the Hawkeyes seem to be making it work, but Hawkeyes not nearly as iconic a character.)