Stop in for a cup of coffee

And you understand that current is flow.
Like water floooooowwwwwwing in a pipe.
:)
Try drawing arrows representing flow and it shows why the ammeter does't show current with engine off, lights on.

Charging-diagram69-mod-ed348-swapbat.png

See?
The current doesn't flow through the ammeter getting to the headlights because the gage is in the alternator output line; and alternator output is zero.

Everything will measure 12 Volts give or take, so a voltmeter isn't very helpful in solving this one. Its either eyeballs and tear everything apart, or eyeballs and an ammeter.