Refresh timing of Google Contacts Sync on iOS?
Your Gmail account is setup to use Fetch, which you have scheduled Every 30 minutes.

Your iPhone will thus sync contacts with your Gmail account every 30 minutes. You may set this to Every 15 minutes if you would like more responsiveness. Note that it may then take up to 15 minutes for your contacts to update, although it could happen much faster. 

New data will be pushed to your iPhone from the server when possible is only an explanation for what Push does, and not a blanket statement for every account you have setup.

As you can see from the list underneath it, only iCloud is configured to use Push updates. Holiday Calendar and Gmail are both configured to use Fetch, which operates on the schedule you select at the bottom.

The reason for this is iOS simply does not support Push with Gmail. In fact as of iOS 10 you can only use IMAP Push with iCloud and Yahoo accounts. Apple has never explained this limitation, which does not exist on macOS. It is possible there was a partnership between Apple and Yahoo, since Steve Jobs made a big deal out of providing Push email with Yahoo during the iPhone’s original reveal.

However, as of  iOS 11 it appears that any IMAP server that supports IDLE (including Gmail) will finally work with Push.