So, whatever is sent by the Android to bring the computer out of sleep mode works just fine from inside and out.
Thus it narrows down to "why does the router's WOL function work (inside and outside) to bring the computer out of shutdown and why the Android app (inside and outside) does not bring the computer out of shutdown"?
Obviously the computer is shutdown but "alert" enough via ethernet card configuration etc. to respond to a wakeup from the router's WOL function. Meaning that the network card is apparently configured correctly (via Device Manager) to listen and to bring the computer out of shutdown when the proper packet is received.
You disabled the router firewalls so what the Android sends should get though to the router. What does the router do with that packet? It knows what to do with its own function generated WOL packet but what about the packet from the Android? Does it port forward that packet and to whom and how. As I understand it, the magic packet includes the MAC address of the computer to be brought out of shutdown
Even if the router knows where to send the Android's packet, is that packet correctly formatted to initiate a computer startup?
Anyway, my "thinking out loud" here sort of leads me to the conclusion that the Android app may not be correctly configured to bring the computer out of shutdown. If "wake up from sleep" is working then the Android at least has the correct information for that process and the router proceeds as expected as well.
What is being sent or perhaps not sent by the Android (and the other tried apps as well) with respect to telling the computer to start up is now the question.
Is there a network card configuration that restricts the network card to only listen to the router and not a wider set of devices; i.e., the Android & apps?
If I go into Device Manager and my network adapter, right-click, select properties I have tabs for Power Management and Advanced.
There is also an Event tab that tells me about events with respect to my network card. Do you see any entries regarding "awakenings" or "starts"?
Under Advanced I have a property = Wake on pattern match and three other WOL/shutdown related properties. Look at "WOL and Shutdown Link Speed".
Here are some explanations regarding the settings:
http://www.mysn.de/driver/Archiv/MB8/9270/Win7/05LAN/README/WIN7/English/AdvProperties.html
The only thing I can suggest at the moment is to look go through your network card's configuration to see if one of the property values is not what it should be or is expected to be.
And what is LAN/WAN app are you using on the Android? I cannot duplicate that part of the puzzle but I would like to see the documentation.