For the original poster, NVR switches and magnetic switches are one and the same. Basically the Start switch engages a high power relay, which has multiple contact sets. One or two sets of contacts control the motor, depending on whether it is configured as single pole switching, or double pole switching. An additional auxiliary set of contacts keeps the relay energised when you release the Start switch, until either the supply power to the system is removed or you press the Stop button. The Stop button breaks the auxiliary relay contact circuit when pressed, causing the relay to de-energise and break the main contacts controlling the motor.
If you fit a remote between the wall socket and the switch on the DC, switching off via the remote is equivalent to a power failure or unplugging at the wall socket, and the DC will not run until you activate it at both the remote and at the DC power switch. This seems to be what you have described in the first part of your message, suggesting that the original switch is in fact an NVR or magnetic switch. It is quite common to have these units installed in machinery without knowing it or without it very obvious. About the only only machinery switches that don't work on that principle are ones that are obviously a toggle style switch with mechanical latching. Anything with a momentary Start push button will use some form of electronic/magnetic latching and operate as an NVR style switch.
IF you wish to operate a device with NVR switching, you have two basic option, bypass the NVR switch and go without the benefits it provides, (mainly ensuring that after a power failure the unit does not start automatically when power is restored, but needs a human command (pushing the Start button) to restart), or alternatively the remote system needs to be installed between the NVR and the motor. This approach can be more difficult to accomplish, but retains the auto start protection offered by the NVR system.
My reading of the modification of the switching followed by early motor failure suggests that the person doing this task may have included/excluded some cables to the motor, leaving a run capacitor out of circuit or leaving the start winding of the motor in circuit permanently, both of which will cause excess current draw by the motor and winding damage leading to failure.
Fundamentally, the description given by fof47 is correct except that with AC systems the voltage changes from positive to negative then back to positive 50 times a second in Australia or 60 times a second on USA. Hence the terms Positive and Negative are meaningless, and the term Active or Line is used to denote the higher potential leg of the supply, and Neutral is used to denote the leg that is at or very near earth potential.
I used to be an engineer, I'm not an engineer any more, but on the really good days I can remember when I was.