I've done a battery pull since 340 came out and my DX will still do this. If I turn it off and then plug it in to the charger, it USED TO just turn the screen on with the battery status showing but not a full boot up. Now it will indeed fully boot up. I have to turn it off again (this is usually done when going to bed) or plug it in then turn off.
Theoretically, I shouldn't have to turn it off at all, of course. I just do some nights.