"MCP" is actually a trade name (from Westinghouse, now Eaton) for what is officially called an Instantaneous Trip (IT) Circuit Breaker. Another common term is "Magnetic Only" or "Mag-Only" circuit breaker. All it means is that the circuit breaker has no thermal trip sensing elements. They can ONLY be used as part of a FACTORY assembled tested and listed assembly, specifically a "Combination Motor Starter". The reason is, as per those NEC sections, it can ONLY be used when the motor starter circuit contains a separate Thermal Over Load Relay (TOL). So the TOL protects the entire circuit against an overload, the MCP protects everything only from a short circuit. And MCP (IT) breaker is NOT UL listed, it is only UL Recognized as a component, because it can ONLY be used as part of a larger assembly and needs another device (ie. the TOL) to make it function correctly in a circuit protection scheme.
YOU can never "decide" whether or not to use an MCP/IT CB, other than indirectly by buying a factory built, tested and listed combo starter that uses one.
Historically, Thermal-Magnetic (TM) Cicuit Breakers used to come with a fixed and sealed Magnetic Trip based on the thermal rating. So if you had a motor starter, it was sometimes a problem to get a magnetic trip that could be set to correctly protect the motor windings. This was especially true of small motors, because the smallest CB you could buy was 15A, but if you had a 1/2HP 480V motor with a FLC of 1.1A, and the mag trip of the 15A CB was 150A (10x the rating), it was too high to protect that motor from a short circuit. So Westinghouse came out with that MCP because it had ADJUSTABLE magnetic trips that could be turned down (or up) to closely match the motor needs.
Since then, many, if not most, TM breaker mfrs have gone to making the mag trips adjustable now, so the real NEED for an IT breaker has actually gone away. Starter mfrs still use them however because for them, it saves money not having to include the thermal trips.
If you saw a motor connected to an IT/MCP breaker with no motor starter and TOL, that would be illegal. But... when the Europeans came on the scene here in North America, they use a breaker that has adjustable THERMAL trips. Technically, that was not something that could be used as a "Circuit Breaker" as defined by UL489 here in North America until relatively recently, but you could use it as a "Manual Motor Starter" under UL508, as long as there was a circuit breaker or fuses somewhere ahead of it. Sometime recently (when I wasn't looking), UL changed to allow adjustable thermal trips and still call it a "Circuit Breaker", as long as it is ONLY used on a motor circuit. Officially, that is now called a Motor Protection Circuit Breaker (MPCB). But in Europe, they call them "Motor Circuit Protectors" because the copyright laws that Westinghouse and Eaton had did not apply there. So you sometimes see MPCBs called out as "MCP"s in EU equipment.
Got that?