Can I backup only my contacts to iCloud?

That is where you need to be careful. You need to find out where all your contacts are. Seems you have them spread between a number of places. You need to get them all put into iCloud, and then sync that to the phone, or get them all together on the phone and save them to iCloud. If you wish to keep those other groups on the phone, then you need to go into Settings, and make sure that iCloud is set as your default location.

Clarification question - are you trying to sync your contacts to iCloud.com or backup your contacts in an iCloud backup?

If syncing, and the only thing you want to sync is contacts, go to Settings/Your Name/iCloud and slide off syncing for everything else. You should be able to sync everything if desired.

Try Settings/Your Name/iCloud and slide off Contacts syncing. Wait a few minutes and then slide syncing on.

Last resort - have you tried signing out of iCloud and then sign back in?

Make sure that you are not using more than one service to sync your contacts. It could be that you have your contacts being synced to your mail account and that is set as the default. That could explain why the contacts are not going to iCloud. Do you use a mail provider that does contact sync, such as Comcast, Gmail, Yahoo, etc.? If you go to your Contacts, do you see anything in the upper left corner of the screen, such such as All Contacts, or All Groups? If so, can you click on that and see what other service is listed there?

You're welcome. Glad you got it all sorted out.