“Cloud fax” is two separate terms that together define the entire term.
In this context, “cloud” is used in the technical term – the Oxford Dictionaries define cloud as
3 (usually the cloud) – Computing
A network of remote servers hosted on the Internet and used to store, manage, and process data in place of local servers or personal computers.‘there’s a rich, complex, shared data store in the cloud’as modifier‘once you are logged in to your cloud storage space, you can upload files to it and share them with others’
“Online fax” and “virtual fax” are very similar to a cloud fax, with online fax being the far more frequently used term. It is possible that if “online fax” was invented later in time, it could have been called a “virtual fax” or “cloud fax.”
“Internet fax” is the broadest term. Any fax transmission that involves the internet is considered an “internet fax” transmission including a cloud fax service. It also includes a fax machine that is partially using VoIP for the communication.
“Email fax” and “web fax” both refer to features used in online, internet, and cloud fax systems.
“VoIP fax” refers to Fax over IP (FoIP), fax’s equivalent of VoIP (Voice over IP, or phone over the internet). It is a type of internet fax, specifically a transmission of sending fax signals over an internet connection.
“e fax” is a brand of online fax service.
“PC fax” usually refers to a computer with fax software that acts as a direct replacement for a fax machine. It connects to a landline phone line (just like a fax).
Fax services were one of the first services to move to the cloud – far before the term “cloud” became a popular term in the early 2010s.
We’ve traced the history of online fax (or a hosted fax service) to MCI Mail’s “Fax Dispatch” service launched in 1988.
The invention of the “online fax” before the word “cloud” (in the technical context) entered popular terminology.