Why does my Goodman furnace shut off after a few minutes?

You're barking up the right tree, actually. That little reservoir is the condensate trap, and on a high efficiency furnace like yours the water has to drain out or the furnace refuses to run. Three blinks on a Goodman control board is the pressure switch code, which ussually means the switch isn't sensing the draft it needs. When that trap fills up or the line clogs, water backs up and trips exactly the symptom you're describing. Runs a few minutes, quits, retries later, never warms the house.

Start cheap and easy. Kill power to the furnace at the switch, then pull that plastic tube off the trap and see if water pours out steadily or just dribbles. If the trap looks gunked up, you can pop it off, rinse it out with warm water or a bit of vinegar, and put it back. Check every bit of that small rubber tubing while you're in there. They get brittle, they get green sludge inside, and they get little dips where water pools and blocks airflow. Blow thru it gently and see if it's clear. Also worth checking since it just got cold: if the condensate drain runs outside or through an unheated area, it can freeze at the end and back the whole thing up. Same symptoms. Some people run the drain line into a floor drain or utility sink instead, which solves that for good.

One more thing, since the switch itself is cheap. Those small hoses dry out and crack right at the ends where they push onto the barbs. Give them a squeeze and a wiggle and reseated them firmly. A cracked hose acts the same as a clogged one. If the trap is clean, the hoses are clear, and it still throws three blinks, then you're likely looking at the pressure switch itself or the inducer, and that's when I'd stop and call someone. But based on what you found with water sitting in that reservoir, I'd put decent money on the drain side first. Ten minutes with a bucket and a wet dry vac might get you heat for free.

Following up on the trap advice because I ran into nearly this exact thing on my parents' 90% unit a couple winters back, and the sneaky part turned out to be the little barbed port where the small sensing hose connects to the inducer housing, not the trap itself. That port had a crusty white plug of dried mineral stuff in it, and no amount of clearing the tubing downstream fixed it until I poked it out with a bent paperclip and blew it clean. Also, once you rinse the trap, put a cup or two of water back into it before you fire up, because some of them need that liquid seal to pass the pressure test properly and will throw the same code if you leave them dry.

That water thing is making me look at my own setup differently now. When mine did the start and quit routine last winter, the culprit turned out to be the panel pinching one of the small tubes when I put it back on after checking the filter. Water had nowhere to go and the whole thing shut itself off. Took me two days to figure out because I never connected the timing to me messing with it. So if you take that front cover off again, worth looking at whether anything is squeezed or bent when it goes back on. Do you have to refill or empty that little reservoir thing yourself, or is it supposed to drain on its own somehow?

One thing that bit me with those little sensing tubes was gravity. Mine had a low spot where the hose sagged between two connection points and it would hold just enough water to fool the switch, then drain out slowly while the furnace sat there faulting. I rerouted it with a couple zip ties so it sloped the whole way down and that was the end of it. If you find yours has extra slack, loop it up and over instead of letting it hang, and keep it off the hot spots on the housing. Costs nothing to try before buying a switch.

Agree with the freezing angle, that one got me two winters ago. My drain ran through the garage and the water would freeze right at the outlet on the outside wall, which backed everything up and shut the burner down with the exact same blink code. I kept resetting it for a week before I figured it out. Now I pour a kettle of warm water down the line when the temps realy drop and it clears in a minute. If yours runs anywhere cold, feel the pipe near the exit, it'll be a chunk of ice you can't miss.