According to your description, the internal voltage regulator of the Arduino Expansion Board might be damaged. The USB power line doesn't need to be regulated because it is a 5V signal, that's why you can power the board that way, but the barrel jack is connected to the regulator's input, and the issue might be there.
First, make sure that your power supplies work. Measure their outputs to check if you are getting the 12V expected. Then, you can try to power the board through the VIN pin header. The VIN pin header is connected to the barrel jack power line. If you can power the board through the VIN header, then there is an issue with the power supply connector, but if you can't power the board, then the root of the issue is the internal voltage regulator.
I recommend you to submit a warranty ticket; the Warranty Team will help you with this issue: Intel Support
I tested the 2A adapter and it was putting out a steady 12.3V so I connected it to VIN and the board would not power. Based on what you say above this is a problem with the internal voltage regulator; I'll submit a warranty ticket.