"Gemini" is defined as servers and client communicating using the "Gemini" protocol (and content is usually in the gemtext format).
Both are part of the Internet, which is defined as an INTERconnection of NETworks, thus a physical worldwide network of computers. Any computer with a public IP (Internet Address) is thus part of the Internet (in your house, is usually your box which is part of the Internet and acting as a gateway for your computer).
So : Internet is mostly an hardware network. You then join different part of the Internet by running adhoc software: Web, Gemini, Gopher, Mail, FTP, Usenet, etc…
Those have very clear and crisp definitions, there absolutely nothing to argue about. A "Web Gemini client" means a browser capable of accessing both Gemini and the Web, which is not the case here. It is a "Gemini browser".