Rafa,
I plugged my Finevu into the cigar lighter and turned the ignition to ACC and let the unit turn on and start recording.
When I turned the ignition to start the Finevu restarted like yours did.
(I also tried with a seezeus GT300, with low internal battery level and it did the same thing)
I should note that the battery level indicator on the Finevu was at one bar at the time. Note sure if this is expected dash cam behavior or not.
The strange part is that the blue led on the Finvu power supply never flickers or blinks during the ACC to Start transition.
EDIT: I just tried again. This time I allowed the Finvu to get to a full 3 bars of stored power in ACC. When I started the car it did not restart.