When you check for voltage, you must check both sides at the same time.
Just because there is 110 on each side does not mean there is 220 going to the pump.
Both hot wires that go to the pressure switch must show 220volts at the same time when checked by a volt meter. Going from one side to ground and then the other side to ground, may show 110v each but may not be 220vac when tested together.
I would use a meter and not the little light thing. If there is a problem with the voltage, the light will not show if it's to low or to high for the pump to run properly.
The splices that you made at the pump.
How did you do it?
The pump may have come with a splicing kit, with a pair of heat shrinks. If it didn't, talk to the people the pump came from, they should be able to get you what you need to do the splice right.