Getting per-user websites working on Mountain Lion server
I also had this problem, and (after only a few hours of head-to-desk impact) found the answer here on AD, from 16 months ago.

Apache 2 `UserDir` problem in OS X Mountain Lion (10.8) Server after upgrade from Lion (10.7)

If you look at /etc/apache2/httpd.conf and your DocumentRoot is /Library/WebServer/Documents/, you might not notice that the error you're getting is about a DocumentRoot with a subtly different name.

/Library/Server/Web/Config/apache2/httpd_server_app.conf refers to serving stuff out of /Library/Server/Web/Data/, which corresponds the server you seem to be running.

You (like me) may be a bit surprised about that. It would've been nice for the Server.app install to drop a helpful README in /etc/apache2.  You wisely tested that you had the correct config directory, but were thwarted -- httpd_server_app.conf #includes a bunch of files from /etc/apache2/extra/.