Why is one of my headlights dim/bright mode reversed?

then take it to the place that fixed it wrong we do not have a clue

They problem showed up after accident. Correct? If so you have in all likely hood smashed wires in harness that runs along bottom of core support. Fixed many issues like this one from front end collisions on this model jeep. I would take back to repair shop and tell them its related to accident, which I can guarantee it is. Reverse lights staying on is the shifter detent cable was knocked out of adjustment. Put it in park, key off, go under car and find black plastic cable that goes to a lever on the trans. In the bracket that holds the cable pull back on the big half round clip, it has a slot in it for a screwdriver. After it is pulled back slide cable back and forth in housing make certain it moves freely. Than grab the lever that is on trans and push lever to rear of car. Push in clip and test that the shifter moves free inside the car. Than set park brake, key on, put in reverse, light should come on, push into park, light should go off.

bad connection or bad earth check all connection or bulb

it is your bcm which is your body control module

The silverstar bulbs while brighter, have a considerably shorter life span than conventional bulbs. If the headlights are dim, it would mean a bad electrical connection most commonly a bad ground. If they are just too dim for your liking and burn at the normal brightness, you might want to consider having auxilliary headlamps installed.

Try turning the headlight control knob clockwise so that the arrow is pointing up. You may only have it in daytime lights mode and not actually in headlight mode.

Take it back to the repairer as they have a guarantee on their collision work.

Yes the screen dims automatically when the lights are on to suit night driving. I am not specifically familiar with the 01 model, however on my 08 model, the display brightness is also manually adjustable via a button located below and to the right of the display. The button is marked with a series of short lines assembled in an arc. Hope this helps.

If the bulbs and relays are good, then the switch is faulty.

Have you checked your fuses and relays? There's a few and not just one to check. The daytime running lights are on another relay than the headlamps.