Backups jobs are successful but cannot see differential backup in the media
This part of code clearly states that you are appending the diff backup 

Set @dt = cast(year(getdate()) as varchar(4)) + @m SET @File = ''D:\SQLBackups\SQLBackups_database\database'' + @dt + ''.BAK'' EXEC master..xp_fileexist @File, @i out IF @i = 1 BEGIN --FILE EXISTS set @strSQL = ''BACKUP DATABASE database TO DISK = N'''''' + @File + '''''' WITH DIFFERENTIAL'' END ELSE BEGIN --FILE NOT EXISTS set @strSQL = ''BACKUP DATABASE database TO DISK = N'''''' + @File + '''''' WITH INIT'' END

Instead of carving out your own solution which clearly is not smart e.g. exclude offline db, take log backups etc is not good .. use Ola's backup solution.

Also, make sure you test your restore plan by actually restoring the dbs and trying a point-in-time recovery where possible.