You can have both. And you can have a fairly fast moving "stable" environment IMO - if you can throw enough testers at it. Any community based OS / distribution will have to rely on community testing, and might take a while to find and fix esoteric bugs.
I think actually the stable refers more to the lack of change rather than the "stability" of the codebase. The platform is stable. It doesn't change often. And when it does it is just to back port a security fix from a newer version of package XYZ etc... Perfect for long lived systems.
/I guess, the 'stable' bit refers not so much to how old the application or the package is, but how well it has been tested. That's probably the best way of describing this. Certainly some bugs get through, even in Debian Woody, but you don't have a trivial package update leaving your system unbootable as a general rule. /