Why does one switch only work when another is on?
If your original wiring schematic is correct black and white on the switch this is a switch loop, or switch leg!
the power goes to the fixture. Then the hot comes back to the switch on the white it is not a neutral.
at this point do not connect all whites together as someone said. The white “always hot” should be marked as a hot the black complete’s the circuit to the outside light. The switch turns on and off this connection!
New wiring would need to be run the new light. You cannot tap a switch leg to provide power for another device!
Be careful accepting assistance on the internet because there are a lot of people that don’t have a clue I hope you have not damaged things wiring etc.
This is a prime example of internet only electricians in a circle !