So, if I'm interpreting you correctly (which this is useful information for anyone wishing to buy games/content for Xbox 360 before the digital store closes)
-Games will appear on your download list if you use the Xbox 360 marketplace pages or of course buy them directly on the Xbox 360.
-Backward compatible games won't (or are inconsistent) appear on the Xbox 360 download list if you buy them on the newer consoles or newer webpages/apps. It's a shame if this is true, because there are some great deals sometimes.
I also understand that you can't add redemption codes on the Xbox 360 anymore for digital or physical gift cards.
Lifetime Sales Predictions
Switch: 161 million (was 73 million, then 96 million, then 113 million, then 125 million, then 144 million, then 151 million, then 156 million)
PS5: 115 million (was 105 million) Xbox Series S/X: 48 million (was 60 million, then 67 million, then 57 million)
PS4: 120 mil (was 100 then 130 million, then 122 million) Xbox One: 51 mil (was 50 then 55 mil)
3DS: 75.5 mil (was 73, then 77 million)
"Let go your earthly tether, enter the void, empty and become wind." - Guru Laghima