Between the switch and the linear actuator on the motor is a relay. A relay is an electrically operated switch.
When you turn on the console switch, it provides power to the magnetic coil in the relay. This coil only draws a small amount of current (Amps). There is a heavy duty switch inside the relay and that magnetic coil pulls that switch closed. The heavy duty switch in the relay can carry the large amount of current required to run the motor to trim your outboard. A relay lets you use a small amount of electrical current from your center console to control a bigger electrical current. This prevents having large wires and high currents in your console.
Some relays have more than one switch internally. The switch configurations inside the relay can be different. Double Pole, Single Pole, Double Throw, Single Throw, and others.
When you flip the switch on your console, the magnetic coil in the relay is energized and the switch inside the relay closes sending power to the trim motor. I suspect your relay contacts have welded shut since your relay seems to be closed even with the wire is disconnected from the console. Have someone put their ear to the relay or use a long screwdriver with the point touching the relay and the handle pressed up against their ear. You want to hear if the switch inside the relay clicks. Turn on power. If the relay contacts are welded shut, no click. If there is a click, the relay is probably okay. Although, a relay can have more than one switch inside. One could be good while another could be bad. Pull the relay from it's base. There is often a wiring diagram printed on the relay showing what each pin does. A continuity tester will tell you if the relay contacts are open or close and the diagram will tell you if they are supposed to be open or closed.
If you have separate relays for tilt and trim or separate relays for up and down, swap relays and see something different happens. When you move the bad relay to a different function, that function should now operate when it isn't wanted.
