A servo is a motor with reduction gears and the control electronics all in a neat package.
Umm - not quite.
A servo is a negative feedback system - in most cases, positional feedback - you tell it where to go (what angle), and the servo moves to that angle, then stops.
In the case of RC servos, they aren't continuous rotation, nor are they typically 360 degree rotation, nor can they rotate past 360 degrees (once, several times, or continuously) - save for a few special cases, like "sail winch servos" and similar.
Most RC servos have a maximum range of about 270 degrees (roughly the same as a potentiometer - because that is what an RC servo uses for position sensing, and that pot is mounted usually as the output shaft - hence the output shaft can only move as much as the pot, which for most pots is about 270 degrees).
Sail winch servos gear down the output to the potentiometer in some manner - so that while the output of the servo can rotate multiple times, the potentiometer is still constrained to a movement from 0 to 270 degrees. So while the output can rotate 360 degrees multiple times, it is still a limited amount of travel.
The misnomer of a "continuous rotation servo" is simply a standard servo which has been modified (by removal of the pot and replacement with resistors, or some other means) so that there is no longer any feedback for position, and so the output shaft can rotate continuously (indefinitely - or until the gears wear out). In effect, they are rendered as miniature gear motors with built in h-bridge circuits for control, and can respond to standard servo control signals.
Real servos, though, they are not - thus, the reason the common "continuous rotation servo" is considered "misnamed" by some of us.
You could have such a thing as a servo which allows 360 degree rotation - and even continuous at that - by making the servo somewhat more intelligent, so that depending on various factors, it might decide that even though the servo is at 10 degrees, and you want it to rotate to 350 degrees, rather than reversing direction (the "short way" - which is what RC servos do typically), it might continue "around the long way"; it would have to be a very different servo, though - one that couldn't use a potentiometer for positioning, but likely an absolute digital encoder or a continuous rotation pot - either of which can be expensive. You won't find these in regular RC servos, but so-called "intelligent or smart digital servos" probably have the ability.
You can see that such a servo can still be called a servo and be continuous rotation - because it would always seek to move to the direction indicated, no matter where it is indicated - and if you wanted it to "continuously rotate" - just advance the position ahead of the rotation...
Ultimately, though, the term "servo" is a shorthand way of saying "servomechanism" - which is basically a device that uses feedback for (typically) positioning. That said, though, any feedback-based system can be called a "servo" system.
For instance, in industry there are things called "servos" which are typically DC motors that don't report position (though they can), but rather "servo" on speed - so basically you set the motor to a particular speed, and regardless of load (up to a limit of course), the motor will maintain that speed (if something tries to slow it down or speed it up, it will compensate to maintain the set speed - either accelerating or braking as needed).
Technically, an engine governor or a thermostat can also be considered a "servo", even though they aren't thought of as such. You could have a light that kept a room at a constant brightness, despite any outside fluctuations in light from the window, sun, moon, and clouds - there again, is a servo system.