I am trying to add a 7 volt light to remember to turn off cellar lights. Wiring white to white or the reverse-either way, the indicator light goes on & off in reverse of the switch. What have I done wrong?
You appear to be connecting your light to the switch, not to the lights. The switch appears to be wired as a switch loop, and the behavior you have seen is exactly what's expected in that case.
A switch loop is an old, but acceptable if it was installed when still acceptable method of wiring a switch using /2 cable with just black, white and ground. The white should be "always hot" and the black should be "switched hot" - the white shiould be marked at both ends with tape, paint or some other method to indicate that it is hot, not neutral.
Power consisting of hot and neutral comes to a light location, and the switch loop connects "switch loop white always hot" to incoming "black always hot." that goes to the switch, where the switch connects it to "switch loop black switched hot" and it returns to the light, where it connects to the fixture hot. Incoming power white neutral connects to fixture neutral.
Your "pilot light" needs to connec to the switched hot and neutral, but there's no neutral wire at your switch location - so your pilot light needs to connect to the switched hot and neutral at the light location.
Wire more lights off a light, not a switch
Electrical wiring does not bring all wires everywhere, that would be a waste of money.
"The wires you need to run a light", are found at a light, at the risk of stating the completely obvious.
You are trying to find them at a switch, and that does not work that way. Bringing a lamp-needed wire (neutral) to a switch would be pointless, at least prior to 2011 when it was mandated to support smart switches.
You cannot misuse safety ground as neutral. If you do, and anything else goes wrong, it can create a fire or lethal situation. Safety ground has one job.