This lack of differentiation really grinds my gears. Why in the world do both of these activities share a name? It would be really interesting to take a random sample of the population and ask them some basic context like their occupation/education, and ask them whether a digital signature comprises a graphic of handwriting (validated with eyeballs) or something more sophisticated (validated with math).
There will be some obvious trends, but I suspect there will also be some surprises.
You're referring to "term overloading". This is pervasive throughout all domains of engineering, but more so in software because there are so many conflicting standards, definitions and citations. It's really hard to get a handle on. Like, I would assume that posting on HN the audience would assume I would not confuse "overlaying pixels of my signature on a document," with, say ECDSA sign & verify. But I was wrong to assume that. So, barring a common definition, should we speak with increased precision thus verbosity? Perhaps. But if THIS example grinds your gears, hooo boy, hang of for a ride.... :)
The point of a signature is to affirm the authenticity of something. When you sign something by hand, you're showing that you reviewed it. If you cryptographically sign something, you're doing the same thing to a bunch of bits, and arguably in a way that's a less easy to forge.
There will be some obvious trends, but I suspect there will also be some surprises.