Charging with Battery Isolator

I have a True Battery Isolator. Question is when you are going to charge both batteries (with the key off), should you connect your charger to the main battery, or the auxilliary battery?

When the engine is running, the blue isolator light is off, indicating the auxilliary batter is charging. When charged to the appropriate level, the light goes out, and the main battery charges.

when I'm hooked to the main battery, key off, and plug in my charger...the isolator,light is off. In approximately 10-15 seconds, the light comes on.

For those with a True Isolator....Is this what you do and see or expect?

Or, do you use two chargers?
Thank you