Computers do not have to be electronic. Counterexamples: Mechanical calculators, LEGO logic gates, hydraulic logic valves, electrical (not electronic) relays. Heck, even human meatbags were called "computers" back in the day.
Watches are nice because it’s much less common to have such tiny precise physical machines anymore, since so many of these use cases have been replaced by “computers”.
It was pretty clear to me that the 3rd sense of "digital" pertains to modern-ish electronic digital computers. I would not call mechanical calculator, human or hydraulic logic "digital computer". (Relay computer is on the fence)