We are on opposite sides of the world.. so it does take a while at times to give you answers.
Also getting old.. and crotchety.
Thanks for the screenshots.
It helps because some of the details can easily get missed.
I don't see any issues in the setup except.. you have set DMZ. Is this required?
DMZ .. apple term is default host.. but I will stick to DMZ.. means all unconnected incoming packets go to 10.0.1.253. That is your server device.. not 10.0.1.6
The port forward goes to 10.0.1.6
I am not sure which has priority. In other words a packet seeking a connection on port 25565 arrives at 47.x.x.x looking to make connection.. Does the Airport DMZ grab it and send it to 10.0.1.253 or does port mapping get it and send to 10.0.1.6
I have not done the test to see.. but in general use either DMZ or port forwarding not both..
What do you have connected at 10.0.1.253.. for example a second router.. that is another cause for trouble.
If that does not work.. just as a test.. and I am not sure how much cable this will take and so might be a request more easily made than accomplished.. plug the server pc directly to the internet connection.. you will need to power cycle the modem for 10-15min .. ensure you have a public IP directly onto the computer and test that the server is working remotely. This should work without problems.. if it does you have some bigger issue with ISP.
Then you need to swap back to current setup.. so a bit of work on your part.. but it ensures everything is working if you bypass the airport and the rest of the gear.
That would be the first thing to try.