“3 wire control” is where you have a Start-Stop control scheme, whether via push buttons or a PLC, held in (“sealed”) by an aux contact of a contactor. The concept is all about “low voltage dropout”, where the contactor will drop out at 70% of the control voltage and not pick up again unless you have 80%, values that are in NEMA design specs and most MIL specs. This prevents unexpected restarting after a power loss; in other words it’s all about safety.
But there is nothing mandating 3 wire control or even how it is accomplished. I once built an entire MCC for some barge mounted pumps that were going to be manually controlled but needed low voltage dropout, using Molded Case Circuit Breakers that were UL listed as Manual Motor Starters had adjustable thermal trips, by adding an Under Voltage Release coil to them. The breaker handle was the On-Off control, the UVR would drop them out under the same conditions as a coil.
There are some OSHA safety requirements that can be met by using 3 wire control, but they can also be met in other ways, this is just the least expensive.
On-Off without a seal-in contact is called “2 wire control” and is fine for loads that you WANT to come back on after a power loss without intervention, like sump pumps and maybe some types of fans. But all aspects of safe operation must be considered.
This all has nothing whatsoever to do with starting current or inrush.