Bob gave a very good explanation. I'll add that the term "enameled" is a
carryover also. I doubt that real enamel is used any more, and that
"enameled" wire is actually coated with plastic these days. But we still
use the term to describe wire, usually solid, with a very thin, usually
transparent, coating. And because it's by far the most commonly used
wire type for winding coils, "enameled wire" and "magnet wire" are often
used interchangeably.

Roy Lewallen, W7EL

Bob wrote:

Magnet wire is a generic term for wire intended for winding coils, with very
thin insulation so as not to waste precious winding space.

Enameled wire is simply one form of it, probably the most popular. Enamel
is a lot thinner insulation than a plastic sheath so won't waste winding
space.

In the old days there were enameled, single cotton covered (SCC), double
cotton covered (DCC), single silk covered, and some others. Today enamel is
used as well as various plastics. One popular type melts away with a
soldering iron, making stripping the end unnecessary, very convenient.

They are all magnet wire.