Thanks. I have an Xbox One on the same account, so I don't know if that makes a difference, but I'll try to see if I can use the other 360. If not, Black Friday is right around the corner, and I might be able to swing another external HD.
Throwing Xbox One in can be a bit weird. If you're playing Xbox One games and not BC 360 games, you can have your main account signed in online on it and a 360 at the same time. If you switch to BC 360 games, then all the stuff in my previous post applies as though you are on an actual 360.
I'm not sure what it does when you play an Xbox Prime game. It might be the 360 rules but since it doesn't actually give you the 360 interface I'm guessing it works like an Xbox One game.
Generally swapping around between consoles isn't too bad if you have internet or multiple accounts you're working with across them. I used to do LAN stuff pretty often so I had to figure out the digital license nonsense pretty well. I'm sort of building up to having a semi-permanent LAN setup for 360 Halo (and Gears and whatever) in my house, but not looking forward to managing 16 accounts in the future to have all the games updated, with DLC downloaded.