Is a free-standing makeup mirror a vanity mirror? Yes. Strictly speaking it is a mirror intended to be placed on the vanity or counter-top, although you'll sometimes see almost any sort of makeup mirror, or even a bathroom wall mirror described as a "vanity mirror". Not by us.
There are lighted and non-lighted vanity mirrors to choose from. Most are reversible, and the lighted group is almost all LED-illuminated. One advantage to a vanity mirror vs. a wall-mounted mirror is the portability of a vanity mirror. Easily move it from room to room, or from "vanity to vanity".