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Isn't the fax machine from the mid 1800s?

"...first commercial telefax service between Paris and Lyon in 1865, some 11 years before the invention of the telephone":

https://en.m.wikipedia.org/wiki/Fax#Wire_transmission

To me it is actually a digitization of an analog process, then back to analog. The tape with the holes punched appears to be digital (holes are either punched or not).

Then a carpet loom is digital too, because it is punch card operated.

You are correct. We're used to thinking of "digital" as implying "electronic", but that's only because our world contains huge numbers of digital electronic systems.

The opposite exists as well, "electronic" is conflated with "digital" quite often when it does not mean that.

> holes are either punched or not

This describes a binary encoding, not a digital process. Otherwise any powerbutton is 'digital', because it's on or off.

You are also correct: power buttons are digital. Contrast to a dimmer switch that can be set to any arbitrary point in a range.

For some interpretation of "low".