SQL Server 2012 backward compatibility for backups with 2008
There's two things involved here: the version number of the file, and the compatibility level.  When you attach a database to a newer major version (like from 2008 to 2008R2, or 2008R2 to 2012) the database version is changed permanently, and you can't attach that database to an older version again.

The compatibility level is for parsing old-school deprecated T-SQL that used to work in older versions of SQL Server.  It doesn't change how the data is stored on disk.

To give the database to someone on an older version of SQL Server, you'll need to export the data and import it into another database.  Tools like Red Gate's Data Compare are helpful for this.