Unplugging Your Xbox: Yes, It Disconnects Xbox Live (Obviously)
Yes. Unplugging your Xbox (power cord) immediately terminates your connection to Xbox Live. No power = no network.
Why It Happens
No Power = No Network Stack: The network interface (Wi-Fi/Ethernet), CPU, and OS all power down.
TCP Sessions Drop: Your authenticated session with Xbox Live servers (matchmaking, party, cloud saves, license validation) is severed.
No "Sleep" on 360/Original: Unlike modern consoles (Instant-On), Xbox 360 and Original Xbox have no low-power network standby. "Off" = Cold boot next time.
What Happens Next
Online Match: You disconnect immediately. Teammates see "Player left." Possible penalty (ban/derank) in ranked modes.
Digital Game Launch: Fails license check (requires periodic online ping).
Cloud Saves: Unsynced progress lost.
Downloads/Updates: Cancelled/corrupted.
Xbox One / Series X|S Nuance
Instant-On Mode: "Off" via button/voice = Low power standby. Network stays alive for updates/downloads. Unplugging power cord still kills it.
Energy Saver Mode: Full shutdown. Same as 360.
TL;DR: Xbox Live requires electricity. Unplugging = Offline.