What could be causing this interference and how to eliminate it?
I've noted that you never mentioned how many power drains device 1 and device 2, in that order of ideas I need to ask first if on the area surrounding there are more devices connected to that branch of AC mains, if there are more devices loadind the circuit/branch it could be happening that when you turn ON the device 2 it has an instant consumption in addition to the consumption of the other devices large enough for the DAQ to detect the instant disminution/change on the power supplied by the power supply (redundancy here) protecting itself and consequently restarting itself. In case of the DAQ doesn't restart itself it could be happening that the DAQ loses some internal electrical reference to realize good measurements but in any case the effect results the same: the DAQ gets upset.

In that case you could try the same operation of start device 2 in the same AC mains where is device 1 but disconnecting the other devices which not need to be tested, if that doesn't works then connect device 1 or device 2 (or both) to other branch of AC mains where one of both are alone, or to a branch where the load is lower than the original branch.

As last comment if there is a fluctuation on AC mains when you start device 2 it probably could be detected or measured with a multimeter set in max-min or by simply watching the screen, if the fluctuation is fast enough so it can not be detected with a multimeter then you'll need a load analizer capable of detecting fluctuations, if the problem is internal to device 1 then you'll need other method like a scope for checking power supervisors internal to device 1 or reference signals.

I hope all this could help you.