Went out to the car to check which light it was and discovered that I was trying to remove the wrong bulb! The one I needed to remove is the dual filament running/brake light. I twisted the plastic socket for that and it came right out without any problem. I removed the bulb and put it back in to the socket and it doesn't work at all any more. I think the problem must have been a filament that was nearly burned out, but not quite. Probably just needs a new bulb.
For future reference, how do the blinker lights come out? White plastic connectors behind the blinker light do not look anything like the socket for the running/brake light. It almost looks like you need some kind of special tool to take them out.
Thanks.