Sometimes the sign is used to indicate which "direction" the temperature is moving. If it was -10° overnight and it's -0° now, the puddles outside will still be frozen. If it was 10° overnight and it's 0° now, the puddles will still be liquid.
(Edit: no idea whether this applies to the Apple watch, it's just a use case for -0 with regards to temperature.)
Your predictions about the state of the puddle are most likely right, but not for the reasons that you think.
Air temperature is commonly measured at 2m above ground. An measurement of 0° air temperature does not imply 0° ground temperature. The more significant effect is that the ground has a higher thermal capacity than the air, so it changes temperature more slowly throughout the day. If it's been colder before and now it's 0°, the ground is still way below 0° and thus puddles are frozen. If it was warmer and now the air has cooled down to 0°, the ground is going to be a bit warmer still and thus puddles are liquid.
Denoting this difference as +0° and -0° does not seem very useful since the same effect is going to be nearly equally significant at 1° or -2°.
(Sidenote: The thermal capacity of the ground is also the reason why air temperature is not measured anywhere near the ground.)
The thermal capacity of the ground is also why, if you're insulating a basement or slab, you want to insulate the edges and DOWN - but you don't need to insulate in the center of the slab (as eventually the ground reaches the ambient temperature and acts as a huge thermal mass).
You're right. What I meant to say was that it's possible the programmers working on the weather app didn't add explicit rounding, they just passed the value through a formatter that did the rounding implicitly.
I believe that's actually a standard presentation, I've seen it in several weather graphs.
It's basically a rounding yeah, signifying it's just slightly below zero, but not enough to round to -1°.
