I don't know much about clocks, but the Swiss railway clocks are by some considered a canonical example of an analog clock, and they behave the way GP describes. But all other analog clocks I've ever seen behave the way you describe.

https://en.wikipedia.org/wiki/Swiss_railway_clock

And conversely there are clocks with smooth movement of the second hand, something I prefer because then there's no annoying ticking noise.

Interesting, the one I have moves the minute hand about 5 times per minute or so, rather than being continuous.

And conversely there are clocks with smooth movement of the second hand, some