Consumers paid money for Xbox Live for dedicated online servers for games. You didn't need to pay for Xbox Live to download dlc.
They changed it over time but at the start that's how they did it.
Did we use the same xbox live? Microsoft did not run dedicated servers for games. They didn't even offer that until the Xbox One
If a game on the 360 had dedicated servers it was on the developers to run and maintain the servers. Halo 2, 3, Reach and 4 all didn't have dedicated servers and were all strictly P2P for all modes, so they weren't even offering it for their own first party titles let alone host them for 3rd party devs.