A relatively simple reason could be causing this albeit that it might require the rear casing to be removed for access, and that is when you push in the charging lead it opens a switch contact attached as part of the socket which disconnects the battery as the power source, likewise the reverse removing the lead should allow it to close which reconnects the battery again as the power source.
Unfortunately on some occasions this might not happen because the spring type contact hasn't made properly (or that the contact has became heavily oxidised) which really requires gaining access to it to bend it slightly, however I would try giving a "quick" squirt of an electrical cleaning fluid into the power socket opening then immediately rotate the entire camcorder a couple of times to allow the fluid to disperse, this followed by pushing / pulling the power plug rapidly in and out of the socket about a dozen times or so to further disperse the fluid.
If the aforementioned still doesnt work then the rear casing will have to be removed for access to the socket / switch.
