Battery light is on so alternator voltage was less that battery voltage, that's what that light means
(Like key on, engine off)
Battery voltage is 13volts MAX, usually 12.8v to 12.3v
Alternator's MIN voltage output at idle is 13.5v, so engine running, alternator voltage will ALWAYS be higher than Battery voltage, unless alternator has failed
So all electrical systems are running off Battery voltage when Battery Light is on, it takes 30-40amps to run a vehicle's electrical system while driving
Anything electrical can/will act up as battery is drained
And Vehicle batteries are not made to be drained
They are made to supply high amps for a few cranks of the starter, then to be recharged quickly
Steady drain can kill the battery, best case shorten its life
Which is why failing alternator can take the battery with it or soon after, so heads up on that
There is an alternator fuse in engine fuse box
And in 1997 Rangers an alternator fusible link on starter relay post
I get my Ranger alternators from wrecking yards, factory ones, they tend to last longer than "new" these days
New alternators and starters just don't seem to last any more