You mentioned that you replaced the line cord to the machine - Did you replace the cord cap (The end that plugs into the wall) or was it a molded end piece that you didn't have to mess with?

If you installed a cord cap, open it back up and make sure you don't have a whisker (single strand) of the wires out of place.

If this is a new cord with a molded cord cap, use a meter on the 20 M setting and measure from white to black, black to green, white to green. You should see high (near infinite) resistance. If not, then disconnect the end of the cord inside the machine and measure again. If you do not see infinite, you got a bad power cord.

Assuming the GFCI is good, then there has to be something wrong with the cord or cord connection. I don't see anything jumping out at me in your pics.

I'm not sure how the filter being wired backwards would cause the GFCI to trip with the machine turned off, unless the filter is bad.

What does your meter read when both leads are together? In other words, what does your meter read when it should be zero?

Also, what does the ground braid to the ground prong at the wall end of the cord read?

Have you disconnected the cord from the filter and verify that the cord itself is good?

What do you read from the ground braid just inside the door, left side of cabinet to the ground prong on your wall plug?

On my machines(Space Invaders, Mata Hari, Power Play, Evel K) I read 0.3 to 0.4 ohms.