Sometimes folks don't realize that in this usage (things wired as in the schematic in prior post) a high impedance coil relay is needed, since they just use matching replacement from manufacturer, or one bought using same part number and supply house provided proper replacement.
But it's simple Ohm's law stuff.
Let assume high impedence lockout relay with 230 ohm coil. And compressor relay with normal coil, with a resistance of 10 ohm. (numbers are imaginary, just being used for example purpose) Control circuit is 24VAC.
In series, that's 240 ohms. 24VAC supply. Amps would equal 0.1. Thus, 23 volts would be dropped across lockout relay coil, and only one volt across compressor relay coil. Lockout will have plenty juice to pull in and stay in, compressor relay isn't even gonna chatter much less pull in.
That's if any of the safeties are tripped.
If they aren't, electrons like path of least resistance. Taking normal control path, with only compressor relay coil in the path (10 ohms), is a lot easier than taking the combined 240 ohms path.
BTW, high impedance relays are also often used in ground fault protection circuits.
I logged in just wanna say thank you to OSIYO for your very detailed and easy-to-understand explanation. Thank you.