I think I see the problem a bit more clearly ... after taking apart a motor starter.
The key thing to understand is that mpotor overloads, while 'switches,' are actually on the neutral side of the coil. They are the only item positioned on the neutral. In this instance, we most certainly 'switch the neutral.'
Mind you, they are also located right next to the coil. Indeed, they are usually physically attached to the contactor assembly.
With that in mind, I have to conclude: ANY connection involving the contactor coils MUST be on the line side of the coil. Therefore, you'd have to use that auxiliary contact to link the starters together.
