Final update for this thread - I called in a ticket to MS, who provided some assistance until the tech noticed we are still on SP3 (there could be a whole other converstion about that and management restrictions on applying service packs). However, I re-installed SQL on my laptop to fix problems with running Profiler, and now my SSMS connection works just fine with Windows Authentication.
The conclusion is that the problem was local to my instance of SSMS, and that the re-install of SQL (and installing SP2 - I control what version is running on my laptop!) was the resolution. What the actual problem was though, I will probably never determine.
Thanks to everyone who provided input.