Other way around. The phone plans were billed by the minute. Then came the Internet. In is first generation, Internet was essentially just a long distance call through a modem. Hence it was billed - like the call - by the minute.

Dedicated Internet wires came much later, and then the dedicated phone lines were dropped as voip was better quality and cheaper compared to the dedicated lines.

While the phone still had a dedicated line it didn't actually need a power connection, as the power through the phone wire was sufficient.