Thank you for the question, was nice and good brain exercise on Sunday 🙂
Got the answer wrong and I kind of tricked myself into it with the INIT as it indicates that "If INIT is specified, any existing backup set on that device is overwritten..." (and RETAIN and EXPIRYDATE is not mentioned, so)
Even though I have noticed the COMPRESSION and NO_COMPRESSION, considering the INIT and it saying I thought both will succeed and but the .BAK file will contain only the 2nd database (as it is going to overwrite). If the default NOINIT was mentioned then the "Restriction" might would have apply.
So here the INIT in the second batch has no significance at all.
I couldn't test the code on my local as it is running on 2K8-R2 Express edition
Msg 1844, Level 16, State 1, Line 3
BACKUP DATABASE WITH COMPRESSION is not supported on Express Edition with Advanced Services.
I see here that the HEADER is untouched when the INIT is used so the information is not entirely erased and it checks to see if the media is been compressed or not.http://blogs.msdn.com/b/sql_pfe_blog/archive/2010/08/09/backup-compression-behavior-when-appending-backups-to-an-existing-media-set.aspx">
The first and the hardest SQL statement I have wrote- "select * from customers" - and I was happy and felt smart.