In the process of diagnosing a plugin issue, I may have accidentally discovered how to detect which server you're logged into.
It seems that each server has its own "game timer" (the value returned by Turbine.Engine.GetGameTime()) which is significantly different from every other server. If you subtract the "game time" from the "local time" (which is the same for all servers) using the expression (Turbine.Engine.GetLocalTime() - Turbine.Engine.GetGameTime()), you get a pretty stable value for each server, that only varies by ±1 second or so:
Server
local time - game time
Gladden
1178127221
Landroval
1178132198
Brandywine
1178171801
Arkenstone
1178716300
Sirannon
1180187358
Laurelin
1180230289
Evernight
1180278158
Belegaer
1180348254
Crickhollow
1283960584
Gwaihir
1286462731
The only question is whether these numbers remain valid after server resets and game updates. I'll recheck them then and report the results here. If the numbers stay the same, it will be trivial to implement a Turbine.Engine.GetServerName() function.
Note that "local time" seems to be a misnomer; the value seems to be independent of timezone. Which makes it handy for this purpose.