Does my Custom Counter have to be connected to the Internet?

Well, YES!

In order to display a number, independently of the option you've chosen (Push number or JSON URL) your Counter needs to be plug into a power source and to an Internet connection (via Ethernet cable or Wi-Fi). 

You can check the following videos in order to connect your Custom Counter to Internet via Wi-Fi: 

Option 1: JSON URL

Option 2: Push number