We've heard them called "Cosmetic Mirrors", "Magnifying Mirrors", "Vanity Mirrors", and most often "Makeup Mirrors with Lights". We call them "Makeup Mirrors" and they comprise all of the other names.
"Hardwired Mirrors" is a sub-category of "Makeup Mirrors" and Hardwired Mirrors are probably responsible for 90% of makeup mirror purchases. A hardwired mirror refers to a makeup mirror with lights that's wall-mounted and connected directly to the electrical system, so there are no visible wires and the lights are always instantly available. 100% of our hardwired mirrors have the on/off switch on the mirror itself, avoiding the necessity of installing a separate wall switch, and the extra expense brought about by that installation.
"Vanity Mirrors" is the term used for free-standing mirrors that sit on the vanity. It cam be the bathroom vanity or any vanity, in any room