Electrical – need an 8 gauge cable for 40 amps at 220v

I'll go ahead and answer because I think your core question is a good one. It sounds like what your company installs are fairly complex home automation systems, and you're hiring residential electricians to do the wiring. Instead, what you need for applications like this is an electrician with industrial automation experience. They do exactly the type of thing you're asking all day long -- wiring rack-mounted PLCs and relays, labeling wires according to ID numbers on a diagram, high voltage and low voltage wiring and termination.

Another thing you need is a contract. If having your circuits and wires marked a certain way is important to you, then you need to specify that in your contract. You can't just assume that your electrician will know that you prefer ID numbers on your circuits just because he has a sheet with ID numbers on it, especially a residential electrician whose usual clients would have no idea what the ID numbers meant. And then, if something isn't done according to what you spelled out in the contract, you can point it out and the electrician will correct it. The other stories you mention just sound like mistakes that should have been corrected by the electrician, assuming they were his error (was it you or he that ordered the relay panel that didn't have enough room for 200 wires?). Anyway, everyone makes mistakes now and then.

To answer your specific question, there already is a system for diagramming electrical circuits. You should probably already be familiar with these symbols if you're installing home automation.

One last piece of advice for good communication with your electrician if you're working in the U.S. (which you may not be): make sure that what you're calling these devices matches up with U.S. terminology for these devices. For example, in the US, you would never call a switch with an "up" and "down" position a pushbutton. Here, a pushbutton is a button that you push into the surface on which it's mounted. Up/down switches are normally just called switches, at least in a house. I don't know what a wallet is, maybe a wall outlet? And it would be very unusual to have a 220V lamp circuit.

In a 208Y/120 4 wire Wye system, you'll get 120 volts between either ungrounded (hot) conductor and the grounded (neutral) conductor. So if you're trying to connect a 120V appliance, all you need is one ungrounded (hot) conductor, one grounded (neutral) conductor, and one equipment grounding conductor.

The ungrounded (hot) conductor will connect to the brass colored screw terminal on devices (the small hole/blade), while the grounded (neutral) conductor will connect to the silver colored terminal (the larger hole/blade). The neutral conductor should never be connected to the equipment grounding conductor, or equipment grounding terminal on devices.

Notes:
Since it sounds like you're a bit over your head, you should probably contact a local licensed Electrician to finish the project and for any future electrical projects.

While you can solder connections in house wiring, it's not a common practice anymore. You'll also want to insure that you are using solder that is rated and labeled for the use, and that your solder joints are strong and insulated.