I think you just wrecked the analogy without realizing it. Your telephone is connected to a telephone network, which is an interconnected network of telephonic devices. Your device is one of many, and has no distinction from others except that it has a unique address in physical space and additionally has an address on the telephone network. Your radio picks up radio signals, a form of wireless mesh network throughout the world (add in signal repeaters used for trunking systems, etc), and you access it with your radio.
In either case of common usage of devices we use to communicate, we don't capitalize the networks, nor do we capitalize the devices (usually, though walkie-talkie I've seen capitalized sometimes).
You use a computer (maybe even the computer - still lowercase - if you have only one in your home) to access the interconnected network of computers to which it is designed to connect via some network adapter and routing device, possibly with a signal modulator/demodulator in there somewhere).
That interconnected network of computers is no different from the interconnected network of radio signals used by trunking systems all over the world, or the interconnected network of telephone signals a POTS line brings to your home.