There's one more step. After you've validated that your new var volume is working, boot into maintenance mode, then double check to make sure that /var isn't mounted, then check again, then when you are sure:
rm -rf *
This will reclaim the space consumed by the old versions of the files in /var when /var was a directory on your / partition rather than a separate file system.
For the truly belt and suspenders, back up everything in var first.