That sounds pretty definitive right there.
I would say you have lost your charging voltage, which led to the battery weakening progressively until it got so slow that it could not start. This led to a flood condition (it keeps spraying fuel even though it is turning too slowly or not at all when the key is turned), resulting in the difficulty starting when under auxiliary power.
With the vehicle finally running again and the battery light lit, this indicates that your charging voltage while running is outside the normal 13-14.5V range..... while it could be an overcharge situation, it is more likely undercharge since an over charge will normally still run when you unhook the cables.
So at that point you are down to an alternator or a PCM (engine computer... it controls the alternator). If you have the ability, I would perform the following tests:
With engine running (jumper cables and all as necessary), measure the voltage across the battery posts, then compare it to the alternator post. If you have charging voltage (13-14.5V) on the alternator but not on the battery, then suspect a bad/loose/corroded connection or cable issue.
If you have low voltage at the alternator too, then check for voltage on pin 2 (red/black wire) of the four pin connector on the alternator. If there is no voltage, expect a blown fuse.
If there is voltage, then you are rightly down to the alternator or PCM. The following inspections will tell you which is at fault:
Measure for voltage between pin 4 (yellow/black) of the alternator 4 pin connector and ground while the engine is running (jumper cables as necessary). You should see around 5V. Slowly add loads (high beams, AC fan, defroster) and watch the voltage.... it should decrease with each load.
Measure for voltage between pin 1 (black/red) and ground you should see very low (100mV etc) voltage. Slowly add loads again and with each load the voltage should increase.
If you see those voltage behaviors, the alternator is bad. If you see one of those wires fixed at 12V or similar unusual behavior, the PCM is most likely at fault (Assuming wiring is intact).
If you wanted to roll the dice.... it is nearly always the alternator. PCM failures for charging malfunction are very rare, I've run into single-digit instances of that in nearly a decade working for Mitsubishi. It is almost always the alternator on this model. That said, do not use aftermarket alternators on this vehicle. The dual-regulator design used by this era (2000+) Mitsubishi is not aftermarket friendly... best case scenario it works for a few days/weeks; worst case scenario (and more common) it blows out the regulator in the PCM, and saving $50-100 on the alternator ends up costing ~$1300 to fix the aftermath. Don't risk it... use genuine Mitsubishi alternators (sourced from a dealership) to avoid any problems.