There's apparently some misconceptions involved here. As for naming conventions, there are established ones and whether you care to use your own instead that's of course entirely up to you. But at least temporarily you should adopt your thinking to the common standard, it makes everything much less of a hassle when trying to determine a problem. When you're trying to find relevant info about anything, if you don't use rely on common terminology it's much more difficult when using your own phrasing and vocabulary. And in this instance it might not even be a 'problem' but more of just getting the initial set up in place.
On a fundamental level, the most generic set up is one already has a Google account, using Gmail for email, the Contacts as your 'address book' (names, email addresses, home addresses, phone numbers, etc.), the Calendar for daily scheduling and such, and other miscellaneous Google services. When initially setting up an Android phone, the user's Google account gets linked to phone, and from that point all those Google services typically get tied to their respective apps on that phone. There's always alternatives and different options but that's the typical configuration. So the online Google account is the focal point, the phone simply gets its info fed to it by the online Google account.
In your instance, the Contacts app on your phone should be synced up with your Google account so all the contact entries you see when using the web interface (
https://mail.google.com) will then also be visible and accessible using the Contacts app on your phone. If for instance you're at home using your computer to read your email and you add a new contact entry, that new contact will then automatically show up on your phone's Contacts app. Conversely, if you're out wherever and add a new contact entry on your phone into the Contacts app, that contact will later then automatically show up the next time you log into your account on your PC at home.
The issue being you apparently don't have this kind of set up in place. If you do go into your Settings >> Accounts >> Google menu, is 'Contacts' shown as synced with your online Google account?
If you don't intentionally want to have your phone syncing with your online Google account, just keep in mind by keeping all your contact entries only on your phone, unless you take the time to make your own offline backups, this is a very, very risky thing to do. Smartphones are most definitely NOT meant to be used for long term storage. Your phone can get lost or stolen, or the operating system might become corrupted, or you might inadvertently delete one or more important contact entries. By keeping you phone in sync with your online Google account if something happens to your phone you still have access to your data. But if instead you opt to only keep things on your phone, you really need set up your own backup solution, and you then need to manually do backups on a frequent and regular schedule.
It does seem a very strange way to label this. You'd think by selecting device, then that's what it would show lol.