My RPi isn't running at present so I didn't look at its /proc/version file, just at the one on this system (running Fedora 22) which IDed both the kernel version and the distro version, so I assumed Raspbian would do the same. My bad.
A way which I know works is to use apt-get to update the system - that shows the version name as part of the stuff that scrolls past.
You could also look at the content of /etc/apt/sources.list It contains a single line. The third item in it is the name of the version you're running. Mine shows (linewrapped):
deb
and I'm running jessie.