Hi there. I don't think the 'Free Unlimited Full Backup' feature is active yet. It's been hinted at and I think found in code, but it's not live, and it only applies (apparently) to pictures taken and uploaded with your Nexus, so even though you may soon have that active for your Google Account, using that account in a different, non-Nexus device might mean those will not be 'free and unlimited' if backed up in full resolution.
Now, as for the Sync option, yes, it must be active for Google Photos to start backing up your pictures. As of now, however, you should only be able to select 'Free' unlimited backup (not in full resolution, as pictures beyond 12MP I believe will be compressed) or Full Quality back up, which would count against your Drive storage quota. Once the feature goes live, you should be able to select the latter option in your Nexus without it counting against your quota, BUT the Sync switch on your phone still needs to be active for the app to automatically back up your pictures. You can, of course, always manually force sync from the app if you choose to keep the 'global' Sync switch off, just don't forget to do it often or you might have a bunch of pictures without a cloud backup.