The only time I've done this so far, the Dom was the narrator, so of course once he'd given her a new name, that was how he thought of her.

In 3rd person it's less obvious, but I'd suggest that it depends on the POV. Even in 3rd person, there's usually a viewpoint character. If it's the Dom, again, he'll think of her as what he calls her. But if it's the sub's viewpoint, you could get quite an interesting sideline on her feelings by going on using her old name for a while, then switching to the slave name at a point where she's starting to accept her new status.