How Do I Know What Mode the Axaet Beacon is in?

When you connect with the 'Eddystone' app it only shows the settings for that mode. That way, you implicitly know the mode.

When you first receive the beacon it is most likely in iBeacon mode.

The number of initial flashes when the beacon changes mode (after fast double press of button) also indicates the mode 1= iBeacon, 2=Eddystone-UID, 3=Eddystone-URL.