There's really not much else to say. When i choose "Run command..." from the K Menu, type "kate" into the "command:" box and hit enter, it runs kwrite instead of kate. Choosing Editors -> Kate from the K Menu runs kate, and choosing Editors -> kWrite runs kwrite. There are also no problems whatsoever, when running them from a konsole.
I'm using KDE 3.5.0.
Problem persists in KDE 3.5.1.
I can neither reproduce this with existing installations nor with a fresh account. Looks like a local misconfiguration.
Deleting the file ~/.local/share/applications/kate.desktop (which was a kde-style *.desktop file pointing to kwrite) solved the problem.