XBox One Green Screen of Death

While writing out a more detailed description of the behavior, I stumbled upon the solution myself.

When you try to install the game over the internet, just let it start, and then cancel it. This will both remove the installation and download from the queue, but apparently also remove the knowledge of the installation from the Xbox One.

So:

1. Start the online install
2. Cancel it
3. Return to the home screen
4. Notice that the disc tile has now changed to "Install disc"
5. Activate the disc tile to start the installation from the disc.

This worked, the game now installs from the disc. Obviously any downloaded DLC's will still need to be downloaded, but for the purpose of this question this is not relevant.

Now, what do you do when you get back home and want to plug in the external drive?

Here's what I did. I first uninstalled the game using the storage management.

This makes my Xbox One forget that I had the game installed. Make sure you don't delete any savegames you have on the console.

Then, you turn off the Xbox One, plug in the external drive, and turn it back on.

Once you do that, the disc tile still says "Install disc". Ignore that. Instead either use your pinned link to the game, the tile for the game (if it shows up on the home screen), or navigate to your games and apps list and activate it. You should get a message saying not to unplug the external drive, and that your game will be ready soon.

A few seconds later you can try again and the game should now start.

Your problem is most likely due to the controller not being assigned to your person.

The other controller, even though it has a headset plugged in is treated as belonging to a guest (or other profile if it is signed in) and, hence, will not have access to the party.

You can fix the issue by pairing the controller with your person manually. However, reading more on the topic, it seems that if you have set up Kinect Sign-in, the console will automatically recognise that you picked up a different controller and will assign it to you as you would expect.

Source: How to assign a profile to an Xbox One Wireless Controller