Yes, it's normal if battery charged enough for system to be started, around 2.9-3v on the battery.
There is multiply Power Cycles, if console aren't charged battery icon will aper when charger connected, and will start automatically if charge will reach "start point".
If battery depleted, charger plugged-in and you will hit power button, system will show logo and because battery doesn't have enough charge to start, system will show battery icon and will not start till you hold the power button for couple of seconds and start it again.
Over all you don't have to worry your system seems to work perfectly fine!