Digital download of Destiny on several Xbox 360 units

in General you are right, it shouldnt matter as both gamertags have a license attached to the Profile.

you can confirm you account can not start the game when your son installed it? i think this might be a glitch

No, this was a free digital download for Destiny on the One if you had already bought the digital version for 360.

From DestinyTheGame.Com

When you buy/redeem the Destiny Digital Base Game, Expansion I or Expansion Pass for Xbox 360 on or before January 15th, 2015, you can download the Xbox One copy from the Xbox Store for free by doing the following:

- Log into your Xbox One using the same Microsoft Account (Xbox Live Account) used to buy/redeem the Destiny Digital Base Game or Expansion Pass on Xbox 360.

- Navigate to the digital content’s location in the Xbox Store.

- Add the item to your download queue and wait for the download to complete.

Expansion Pass content (which includes Expansions I and II) will only be available for full download once released.

From what that says I am guessing you had to download it digitally to the X360.

Preloading worked for me, far as I could tell, for Destiny. The game was pre-loaded onto my console from a previous digital purchase. When the 360 to Xbox One free upgrade deal was announced I cancelled the digital purchase via Xbox Support, but didn't delete
the pre-loaded content. On Destiny launch day I bought it on the 360's digital marketplace, then the Xbox One activated and installed the pre-download that I had sitting on the console.

A download isn't an installation. Unlike the 360, once you download a title it then has to be installed. Installation shouldn't take too long relative to the download time for 44GB on most consumer ISP connections.

MY GUARDIANS, STATS AND GEAR

Q. Will my Guardians, progress, stats, and gear carry over when I download and play the next-gen version?

Yes. Whether you participate in this program or not, your Guardians, progress, stats, and gear are shared within a console family (PlayStation®3<->PlayStation®4 or Xbox 360<->Xbox One) and will be waiting for you when you transition to next-gen.

Q. When I buy the Destiny Limited Edition or Destiny Ghost Edition on Xbox 360, will I also be able to download the Xbox One version of the Expansion Pass that comes with it for free?

Yes, when you buy the Destiny Limited Edition or Destiny Ghost Edition on Xbox 360 and redeem the Expansion Pass by January 15, 2015, you will also be able to download it on Xbox One by doing the following:

- Log into your Xbox 360 and redeem the token for the Destiny Expansion Pass on Xbox Live on or before January 15, 2015.

- Log into your Xbox One using the same Microsoft Account (Xbox Live Account) used to redeem the Destiny Expansion Pass on Xbox 360.

- Navigate to the Expansion Pass within the Destiny content page in the Xbox Store.

- Add the item to your download queue and wait for the download to complete.

A physical or digital Xbox One version of the Destiny game will be required to play this content on Xbox One, sold separately.

Expansion Pass content (which includes Expansions I and II) will only be available for full download once released.

www.destinythegame.com/digital-upgrade