The above responses about cover it all, although I would do things my own way as much from habit as anything.
TRUE: The breaker protects the circuit, not the load. The load will have an "over-current" device somewhere on board. Consider not just the sewing machine above, but any lamp or kitcher counter load plugged into a 15A(or 20A) circuit. The cord alone should be a give-away. Many such appliances, fans, blenders, etc. have what is called "impedence protection" for light loads. If you really want to know what that means, look into
http://www.hudsontelcom.com/uploads/ShopElex.pdf. But that's a loonnngggg lesson.
The reason for this post is to cover specifically motor protection, usually a motor starter VS a contactor. A contactor is nothing more than a high current relay. Sometimes they are "definate purpose contactors" but still just a large relay. Currents of several hundreds of Amps, perhaps even a few thousand. My industrial background coming into play here. . .
A motor "Starter" on the other hand is a "contactor"
with an overload device. That is likely the protection for a mill larger than the small "bench-top" machines. It most likely be a "solder pot" device, although there are other methods.
The solder pot provides a time rating because motors draw enormous currnnt for a few milliseconds when starting. Sometimes 6 or more times run current. The functional element is not really solder but a similar "eutectic" alloy that allows the motor to draw that enormous current for a couple of seconds before it melts. When it does melt, a spring loaded device opens a contact in the motor circuit, dropping out the "contactor" portion of the starter.
Depending on manufacturer, and age, and a few other variables, the solder pot may be directly a part of the heater or may be an external part. The heater is specified very specifically for the motor. Look to the motor nameplate for "Full Load Amps" or "FLA". The heater
must match that number quite closely, just installing "any old heater" will cause trouble. It must be fairly accurate.
There is one exception to this. A motor rated for 10 Amps FLA (for example) might have a heater rated for 8.5 amps. This would be more common for home shop machinists where the maximum load might never be reached. It is advised that the proper heaters be kept close by "just in case" they are needed. The other way, where the heater is larger than the motor, must
never be allowed. That would allow the majic smoke to get out. And we all know how difficult that smoke is to get re-inserted.
The "line cord" must be large enough for for all motors and control circuits. But it is not necessary for it to be large enough for starting current. That lasts only a few seconds and the starter overloads will shut down the motor. Repeated attempts to start the motor in a short time span may well overheat the heaters. The only solution to that is to have a cup of coffee (or adult beverage) and let things cool down. The line cord may get warm but supposedly is rated for that. I'm "old school" and don't trust plastics but the cord should have a temperature rating embrossed on it.