Does a whole-server backup guarantee a reliable backup of running databases?
For a filesystem backup to be consistent MySQL requires that the DB be stopped http://dev.mysql.com/doc/mysql-backup-excerpt/5.0/en/backup-methods.html but as you said there are ways of using the DB provided tools to create consistent online backups.

Other DB engines such as Oracle have a mode you can put the DB files into prior to a known time of a filesystem copy to ensure that what's on disk at that time is consistent but you have to know the exact time the backup will start and finish.

there are other DB formats that are also consistent by virtue of the fact that their files are 100% append only such as CouchDB.

But for the most part backups of a running DB require some kind of intervention using the DB engine itself.