what's an example of an internet? (lowercase i)

[Kaido]

I'm studying for my CCNA exam and I'm learning about network localities (intranet, extranet, Internet, and internet). While gives a brief overview of the differences between an internet and the Internet, it doesn't give any examples. Can an internet be accessed over the Internet? For example, my local library has their book catalog available both in their LAN inside the library and through their website externally...or is this the Internet? Would an internet be more like a dial-up bulletin board that's available over a network connection but not the Internet? I'm a little confused and the book doesn't go into further detail with examples. If someone could clear this up for me I'd appreciate it!