Q: Which contacts are backed up

icloud only backs up iphone contacts and ideally they should not exist on the phone. Ideally only icloud or email contacts should be on the phone and in that case they are not backed up, but synchronized which is way better.

The backup contains the contacts as they appear in the Contacts App.  (They are backed up as well as synced with Contacts in iCloud.) 

The backup also includes email addresses and phone numbers you previously sent/called which aren't in your Contacts and show in the Recent call list or show when entering an address for a new email.

Your iOS device backup only includes data and settings stored on your device. It doesn’t include data already stored in iCloud. That includes contacts, calendars, bookmarks, mail messages, notes, shared photo albums, iCloud Photo Library, My Photo Stream, and documents you store in iCloud Drive.

Source.

iCloud: iCloud storage and backup overview

You are overlooking the complete backup to iTunes, which does include those items.  The OP didn't define the type of backup and the email addresses and phone numbers not in Contacts on the phone are not synced to anything. 

iTunes backups

From your Mac or PC, you can make a backup of your device in iTunes. Syncing your device with your computer isn't the same as making a backup.

An iTunes backup includes nearly all of your device's data and settings. An iTunes backup doesn't include:

• Content from the iTunes and App Stores, or PDFs downloaded directly to iBooks (You can back up this content using Transfer Purchases in iTunes.)
• Content synced from iTunes, like imported MP3s or CDs, videos, books, and photos
• Photos already stored in the cloud, like My Photo Stream, and iCloud Photo Library
• Touch ID settings
• Apple Pay information and settings
• Activity, Health, and Keychain data (To back up this content, you'll need to use Encrypted Backup in iTunes.)

Choosing an iOS backup method (Should I use iTunes or iCloud to back up my iOS device?)

According to Apple, sorry if your opinion differs content that is being synced is not backed up.

For icloud backup it is content being synced with icloud that is not backed up.

For itunes backup according to your own post it is content synced with itunes or stored in icloud.

"Your iOS device backup only includes data and settings stored on your device. It doesn’t include data already stored in iCloud. That includes contacts, calendars, bookmarks, mail messages, notes, shared photo albums, iCloud Photo Library, My Photo Stream, and documents you store in iCloud Drive."

• "Content synced from iTunes, like imported MP3s or CDs, videos, books, and photos
• Photos already stored in the cloud, like My Photo Stream, and iCloud Photo Library"