Online fax is a fax that is hosted by a fax server.
It is a “fax” that is available “online.”
While each of these terms is relatively interchangeable, they each have specific meanings that make them slightly different.
“Internet fax” applies to all types of fax transmissions that even partially use the internet – for example, two fax machines communicating on the T.38 standard are using “internet fax” as part of their communication. The term “internet fax” is the broadest term available for faxing through the internet. In comparison, online fax refers to a type of internet fax technology.
“Email fax” and “web fax” apply to only one feature of online fax. They are both ways that an online fax can be accessed (through email and a web portal, respectively).
“VoIP fax” is another term for FoIP (Fax over IP), a technology that is similar to Voice over IP (VoIP). With FoIP (or VoIP fax), a standalone fax machine connects to the internet for communication with other fax machines over FoIP. This process is different from an online fax, which is a term for the entire system.
“e fax” is a trademark of an online fax company.
A “PC fax” refers to a PC (often a Microsoft Windows computer) that is acting as a fax machine. With PC faxing, usually, there is a need for faxing software as well as a landline phone connection and fax modem.
“virtual fax” and “cloud fax” are both similar to online fax. If “online fax” was invented later, it could have been called one of these terms. However, the term “online fax” dates back to 1990, before “virtual” and “cloud” entered the popular language.
There is no clear history of the term “online fax.”
According to a Google Ngram search, the first use of the term dates to 1990.
A search of Google Books quotes a book called “Fax for Libraries” by Patrick R. Dewey published in 1990 with a section called “Online fax services” as “MCI Mail makes fax available through its Fax Dispatch Service.
MCI mail was an early email provider and provided fax to email through Fax Dispatch, launched in 1988.