English: when to capitalize internet

Tutoring English, proper names come up. The tutor mentions Internet.

The Internet means the whole global network: from your browser, connected to your service provider, you (theoretically) have access to the Internet. An internet is a local network, created and managed by a specific entity for its own purposes. An internet may be self-contained, with controlled access, or even none at all, to the Internet.

Source:

Forouzan, Behrouz A. Data Communications and Networking,, fifth ed. New York: McGraw-Hill, 2013.