When run as a user, Xorg places log files by default in $XDG_DATA_HOME/xorg/. Also, when run as user, -logfile option cannot be used to specify a different location. The file name does not include the host name. The problem with this choice is that $XDG_DATA_HOME can be an nfs-mounted home directory. When a user logs in on multiple hosts using xinit, they all attempt to place log files of the same name in the same directory. The user has no option that can be fed to xinit to change the log file location. A better choice for a default would be $XDG_RUNTIME_DIR/xorg/. The runtime directory us per-host just as /var/log, and is user specific and user accessible.
$XDG_RUNTIME_DIR isn't necessarily persistent though, e.g. on this system it's /run/user/$UID, which is a tmpfs filesystem.
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