How to correctly discard old backups?
My best recommendation for you is to use already applauded backup solution from Ola

CleanupTime
Specify the time, in hours, after which the backup files are deleted. If no time is specified, then no backup files are deleted.

I normally, use backup database ... with init and then delete the backup files older than X days.
Ola's solution is a one stop solution and its very flexible, so you can tune it as per your needs.