No, because the COM terminal of switch 2 comes from neutral according to your sketch so that wouldn't make sense.
I'm a little confused here. How were you turning the machine on and off? By means of the drum switch? Or by switch 1?
I can't tell you where the mystery wire goes because I don't know the terminal arrangement of the relay. That may require some detective work on your part. I'm also not real clear on the pinout of the drum switch as far as the relay coil switching goes. We really need the truth table for both the relay and the drum switch then we could get somewhere; you would need to label the wires, pull them all off, and then use the ohmmeter to id the terminals. It's a time consuming job but it's the sure-fire way to do it.
M