Add a contact to a list?

If you go to Settings>[Your name]>iCloud. Then where it says Saved to iCloud, tap on See All. Then check to see if Contacts is turned on for Contacts. It should be for you to use iCloud contacts. Then go back to Contacts and tap on the iCloud list. Try adding another contact and see if it saves to iCloud. I cannot see any place to select the particular list to add contacts to, but I only have iCloud under Lists. I don't even have All iPhone.

I'll try adding a list and see if I can get Contacts to force me to choose where to save.

EDIT: When I added a new list and went to that list, it shows no contacts. When I tap the + in the corner to add a Contact, I have the choice to move one from iCloud, or add a new one. I think if you make sure that iCloud contacts is turned on and you have iCloud selected for Contacts, when you try to add a new contact it should go to iCloud. Then you probably can delete the All iPhone list by swiping over the name to the left.

When you open the Contacts app on your iPhone, what do you see in the upper left corner of the screen? There should be something there showing <Lists. If you tap on that, what do you see? Maybe either on iPhone, or iCloud, or something else? This is where list management is done. You need to determine where your contacts are being stored first to figure out how to manage them.