I have the same problem, just adding another voice. iPhone 5s 16GB. Verizon. OTA update to iOS 8.1 > I'm showing "Extended" (instead of "Verizon") as carrier/network ever since. My home and office are within a 3 mile radius, in the middle of town, in which I always have "Verizon LTE" next to my signal indicator. I have not left this same radius during or since the update. I have a colleague whose Verizon Wireless iPhone 5s, updated to iOS 8.1, correctly shows "Verizon".
Steps I've taken to troubleshoot (all unsuccessful): Turn WiFi off. Turn Bluetooth off. Turn LTE off. Turn Cellular off. All combinations of roaming settings enabled/disabled. Turn Airplane mode on. Soft reset (power cycle). Hard reset (Lock button+Home button power cycle). Pulled SIM card. "Reset Network Settings". Again, none of these things resolved the issue.
Chat session with Verizon Wireless Technical Support was futile and a nonsensical rabbit hole. I was immediately advised to backup > restore, "but I wouldn't reinstall iOS 8 until a new update comes out". I pointed out that those two statements were contradictory but that did not seem to phase the rep. Furthermore, I was told I would lose my text messages -- when I quoted the iTunes prompt that says "This will restore only the contacts, calendars, notes, text messages, and settings, not the iPhone firmware", the rep confidently replied, "Right. You are protected by the network." Wut.