You don't "need" an indicator light. It is NOT necessary if the wire from no. 1 alternator goes to a DIFFERENT terminal on the ignition switch than where the battery hot wire is connected.
You can have a light if you want it wired as you have shown.
Don't get confused with a common point for now. I realize you are new to wiring so I am keeping it simple. Just wire it and start motor and test alternator output as I described.
NO WIRES TO the old Voltage regulator.Can you PLEASE confirm you have a 12V battery.
Can you post good pictures of your ignition switch and it's terminals or make a drawing.
Be sure to tell us what the alternator output voltage is,, don't just say "no sign of life".
If you need a new alternator
THIS is the one you need.