I believe I have a Gen 3 T-stat, however it looks identical to the photos here. Anyway, the connector does absolutely pull UP, not out. If you look carefully, there are 2 nibs to prevent it from pulling out sideways. To clear the nibs, you MUST pull it UP.
Lastly - THANK YOU for this write-up. I manhandled my Nest last night, and it popped off the base and hit the wood floor. It refused to boot after that. No screen at all, just a flashing yellow light at the top. I tried a lot of troubleshooting stuff, but finally concluded something HAD to have come loose from the impact. Following the directions here, I was able to disassemble it far enough to see the red battery wire had slipped slightly out of the connector… just enough to see copper between the wire insulation and the connector. I pushed it back in, and secured it with a tiny bit of hot-melt glue. Put it all back together, and plugged it into the wall. It booted right up, with the full display as normal.
