Downloaded game on home Xbox with sons gamertag but can't play on second xbox

Hi, unfortunately you can't share between 3 consoles. Best you can do is have one sons Xbox as your home and be signed in on the other sons. Means you can't play whilst your signed into second sons Xbox.

You only need one live subscription on an xbox one as long as the profile that has the live subscription has that console set as their home console.

now to answer your question about your other sons account it sounds like your second son may have an offline profile on the xbox 360 here is a link on how to turn an offline profile into an online profile so that you can sign in with your second sons account
on the xbox one.

this is the link for turning your second sons account into an online profile: https://support.xbox.com/en-US/my-account/manage-gamertag-and-profile/offline-profile-to-gamertag

this is the link that will tell you about the home xbox and how it works: https://support.xbox.com/en-US/games/game-setup/my-home-xbox

Let me try to clarify.

Each digital game you purchase comes with two licenses. One license attaches to your Gamertag, the other attaches to the Home Xbox of your Gamertag.

I have an Xbox upstairs in the kid's playroom set as my Home Xbox for my Gamertag. Anyone can sign in to the Xbox and play my downloaded games and use the Gold multiplayer access. I have to sign in to the Xbox in order to download/install games from my
library for others to play. So if my son signs in to that Xbox and I have Destiny 2 installed, then he can play Destiny 2. He can play with me if I am downstairs playing Destiny 2 on my Office Xbox.

My Office Xbox is not set as a Home Xbox to anyone's gamertag. I sign in to it and can install/play any games from my Library. If anyone else signs into, then they have to play games from their own library.

When I buy a game, I use my account to purchase it. If my son buys a game on his account, then he can install/play it on the Xbox in the playroom but no one else will be able to play it. For example, if he buys NHL 20 then he can install and play it upstairs
but if I go up there I will not be able to even though it is my Home Xbox. That NHL 20 is attached to his Gamertag and his Home Xbox which he has not set yet.

Now if I let o my son log in to my Xbox in the office... nevermind, not going to happen. But if somehow he did and set it as his Home Xbox, then I would be able to play NHL 20 in my office on his Home Xbox while he goes upstairs and plays it using his Gamertag
on that Xbox.

That's why I am confused. His is the home console.

Try having each other as Home Xbox.