"Phantom" touches on appliance's capacitive buttons. A serious problem?
It means that the exercise machine wasn't designed for good conducted noise immunity.  There are lots of crappy designs out there.

The first device is probably putting noise onto the power line, which confuses the second device to where it thinks buttons got touched.

The signal to noise ratio of capacitive sense buttons is pretty small to begin with.  Of course there are ways to mitigate this, but the designers would first have to care, then know what they are doing.

Whether this is a problem only you can say.  We have no way of knowing how important it is to you that the second device doesn't start doing random things when the first device is in use.

As for the washing machine, don't do that.  The plug was apparently polarized for a reason.  I can't imagine why you'd deliberately flip the plug, but next time give the Darwin Awards people a heads up so they can get a trophy ready.