I think your getting it wrong. A network is abstract, right? A network refers to connections b/w a group of two or more devices. The Internet is a network itself composed of nested networks and then end-user devices.
So, IoT means the network of all things, e.g. the Internet of all things.
The phrase "connected to the Internet" infact should be corrected. But the fact is, alternative phrases don't exist. Your phone doesn't connect to the internet, rather it becomes a part of it