How contacts are supposed to work in the Outlook web app

Add to contacts is shown when the contact is not in your contacts folder - usually from the recipient cache AKA autocomplete list (or if you previously used Outlook customer manager - the custom list still exists hidden in the mailbox). I also see it on some contacts moved into that mailbox from a different account or that I added on my cell phone but it's possible they are in the autocomplete list and is showing that entry.

Outlook on the web hides duplicates, so you might only see 1 - the autocomplete entry. 

New Outlook app and classic Outlook only show the contacts in the folder. 

FWIW, you'll see similar weirdness in classic Outlook when you Search people. If the entry has Add to contacts, it's from the autocomplete cache. 

Contact lists is for contact groups / DLs.  The web separates them from contacts.