An internet is simply a connection between two networks.

Huh? That's what I would refer to as a WAN connnection. Any time you link two networks together you CAN call it an internet - but sod that, it's too confusing. A group of computers that are linked ('networked') together, e.g. using copper cabling and a distributor, i.e. switch or a hub is a network - agreed?
More precisely, if the network is confined to one geographic area then it's a L.A.N. (Local Area Network).
If the communications network spans a wide geographic area, such as a country or continent (using routers), then it is a W.A.N. (wide area network). I don't know many computing specialists who would call a WAN their 'internet' -After all, one can't hear a capital :-)
Anyway, LANs can use many different technologies to connect to remote LANs and form a WAN, e.g. dedicated lines through an ISP ($), tunnelling and VPNs (using the Internet), Frame Relay - and lots of things not worth going into here.
Generally, most refer to the internal (private) network of a firm (be it a LAN or a WAN) as the network or intranet. Some understand intranet merely to encompass the firm's internal website though.
As one might imagine, definitions of the Internet vary greatly, and it's all very complicated and abstract anyway. Something like this will do though:

A decentralised global network connecting millions of computers using countless services and connection methods provided by an ever-growing IP protocol suite.