If you get the message: "No backups available" it is because there is no iCloud Backup in your iCloud that you can restore from. If you can see your photos, contacts and notes on iCloud.com, it is because the various things are synchronized with iCloud. This is good because then you just sign in with your Apple ID and photos, contacts, notes, etc. will automatically sync back to your new iPhone.
Regardless, an iCloud Backup does not back up the things that are already synchronized in iCloud, so a backup will restore the organization of your apps on your home screen, as well as settings. Everything else that is already synchronized with iCloud is not saved in an iCloud Backup.
So! When you set up your iPhone, select "Don't Transfer App & Data" and then sign in with your Apple ID. Your iPhone should download all your content back to your iPhone. However, you must set up the settings yourself again, as well as download your apps. To do this, long press the App Store icon to bring up a pop-up menu and select Purchased, and you'll get a complete list of all your previous free and purchased apps that you can download again.