Hopefully there's just simply some glitch in the syncing process between your Google account and the Contacts app on your phone.
Try this, just to be sure, go back into
https://contacts.google.com and confirm all your contact entries are there. In the left side panel click on 'More' and in the resulting drop-down menu there should should some additional items. Select 'Export' to go through a process of exporting your contacts, so you have a backup, just in case. (Backups are important even when everything is working right.)
Now go into your phone's Settings >> Device >> Apps menu, scroll down to find the 'Contacts' app, and open it. Tap on the 'Force Stop' button, then tap on 'Storage', and then 'Clear data'. Using 'Clear data' will wipe all its previous settings, essentially returning the app to its first time use. Now go to Settings >> Personal >> Accounts, open it, tap on 'Google' and confirm that 'Contacts' is enabled. Close all, go back to your Home screen and swipe up to access the apps menu, and tap on 'Contacts' to open the app up again. It should be trying to repopulate itself with your contact entries as it re-syncs to your Google account. If it's not, try tapping on the three vertical dots in the upper right, open up 'Manage accounts', and tap on those three vertical dots in the upper right and confirm the 'Auto-sync data' box is enabled.