PBX and Internet Phone

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Initially, it was private since in general it served a user company, which wanted to have its own branch to save some money in, for example, internal calls. This was done by having the exchanging or switching of circuits done locally, inside the company. Because fax machines, modems  and many other communication devices can be connected to a PBX, extension
developed to a generic term describing
all devices connected to a PBX.
The PBX's are connected to the outside world by a number of lines called trunk lines.After the PBX business took off, PBX's started offering services to small businesses and home users that were not available in the operator network. The status quo was changed after several decades by two significant developments during the 1990's. One was the massive growth of data networks  and the other one was the trend of focusing on your core competence. Data networks meant that companies had to have packet switched networks anyway, so putting the telephone calls there was tempting as it was. The Internet - and its low price on global communications - was the final straw, and the VoIP PBX was created. Technically nothing was being exchanged anymore, but the term PBX was very well recognized, and there apparently were no major pushes for a different acronym.

On the other hand, most companies realized that handling their own telephony was in fact not their core competence. Not to mention, PBX services had always been hard to arrange for smaller companies. These combined to create the concept of Hosted PBX, where the PBX was actually managed by the telephone company, and the customer company just had to sign up for a service rather than buy an expensive piece of hardware and then maintain it. This essentially removed the branch from the private premises, making it centrally managed. Some simply gave up the PBX for Centrex service.