Never remove a battery terminal with the engine running. Unless you want to permanently disable the alternator.
Don't Disconnect the Battery with the Engine Running
I'm pretty sure the low output mode is still charging. I can drive as far as I want without losing any voltage. Honda says 12.5v, but my multimeter tells me it's more like 13v. The easy test to see if it's charging or not is to force the alt in the "low" mode and drive with all the electrical accessories on while watching an accurate multimeter. If the voltage remains steady, the alt is charging. This is what I'm seeing in my '91 Civic.