With many types of relay coils, 120 volts IS enough to ‘hold’ a 240 volt coil, but not enough to actually engage it. To put it technically, 120 volts may be above the dropout voltage of the coil but below its pickup voltage.
There are three voltages you talk about when discussing potential relays:
The PICKUP voltage is the lowest voltage which fires or engages the contacts from the resting position; it is lower than the HOLDING voltage.
The HOLDING voltage is the highest voltage you can continually apply to the relay coil without overheating it.
The DROPOUT voltage is that voltage below which an engaged contact returns to the resting position; it is always lower than the PICKUP voltage.
You can take an ordinary contactor and make it into a functional potential relay by wiring an appropriately sized resistor (sized properly for resistance AND power dissipation) in series with the contactor coil. Or you can use a potentiometer. Either way, you can tailor the coil to have the pickup and dropout voltages you desire to do a given task.
In order to do the resistor calculations, you need to know the coil current or its impedance or equivalent series resistance (ESR), then by Ohm’s law you can calculate the resistance and power dissipation for the resistor to be placed in series with the contactor coil. Often this involves considerable trial and error, but you can always get where you want to be.
So, how might this actually look in a shop? Let’s say you did the work above to find the appropriate resistor to wire in series with the ‘output’ contactor coil. You wire it in series, and energize the contactor coil with, say, L2 and L3 (generated leg). In addition, you place some kind of toggle switch in series with the coil energizing circuit, so that you have a way to manually shut off the distribution panel even if there is no phase loss on any of the three phase legs.
I think you have it! If that distribution toggle switch is ON, but L3 fails, the panel is dead; ONLY if BOTH the toggle switch is ON, AND the generated leg is ‘good’, do your three phase machines get their power.