“If” the earth ground is a separate ground connection (only grounds the case) from the signal ground, you want to use the signal ground.
Here in Canada on any appliance that has a three prong plug, that third prong goes to ground, earth ground. And in the fuse box, the neutral and ground are connected together. You can debate if that’s good or bad in reference to a lightning strike but it is what it is. I personally don’t think it will matter, your going to have dead tech anyway.
If you have an ohm meter you can check to see if the earth ground and signal ground are one and the same. They usually are. If they are the same (wired together) it doesn’t matter which one you use.
If two pieces of equipment are a fair distance apart, you can have different earth ground potentials. I do believe its referred to as a ground loop. Something like that. It’s 6AM here and I only just got up. Haven’t had my fist cup of java.
Anyway, if you just reference to earth ground and only run the one signal wire, you can end up with issues. Run a second wire linking signal ground from both devices and things will be much better. They have a common reference that doesn’t change.
The earth ground is IMHO, there to protect you, not the equipment. Especially with mains AC powered devices. If something goes wrong inside and the live feed contacts something its not supposed to, it gets shorted to ground and trips the breaker or blows the fuse.
If I got something wrong there feel free to correct me. I’ve been out of the loop (retired) for a fair number of years and my memory isn’t what it used to be. ;)