Single-Word Requests – How to Describe a Thin Mist Rising from a River?

We would identify something as a river because we ordinarily expect it to have the characteristics of a river, even if some of those characteristics are temporarily suspended. That is, a permanent or semi-permanent stream would be a river even if it has temporarily stopped streaming, as noted in the comments, by drying up, being dammed up, or being frozen solid.

On the other hand, something which only temporarily takes on the characteristics of a river could be called so, metaphorically, but only while it possesses those characteristics. If they cease, the metaphor is no longer applicable. If I spill a jar of molasses, I may report a river of molasses running down the table leg, but once the flow stops, we would lose the river (having gained a pool).

Consider one's vocation. Someone who introduces herself as a pilot or a writer does not cease to be when she is not engaged in the act of piloting or writing. On the other hand, if I am speaking with my friend about two women across the room, one dancing and the other standing still, I might refer to the first as the dancer to distinguish between them. If she stops dancing, then she is no longer a dancer, and I will have to identify her otherwise-- unless she hands me her business card with "dancer" given as her occupation.

The three have similar properties but have different size scales.

Haze is large scale like smog that you can see over long distances like obscuring a skyline or the horizon. I associate it with warmer dryer weather. It is very vague.

Fog is scientifically a low lying cloud, but when you're inside it, things are just not as bright and you can't see as far depending on how thick it is, and things feel a little cool and humid at the same time. There's very little differentiation inside, it's just heterogeneously a little less bright, and you can't see as far and it looks whiter in the distance, but there are no noticeable cloudy areas. Outside of it like from a mountain top you really do see that it is a cloud.

Mist is to me a kind of rain, or barely rain at all just a noticeable wetness in the air that an umbrella just really doesn't do anything to stop. You don't really see it falling but standing out in it, you eventually get wet.

Also, since language isn't logically consistent, a 'mister' outputs mist which looks like a small scale fog, a fog machine outputs very small scale fog that is hardly at all wet.

Hazy, foggy, and misty tend to follow all these but not always perfectly and usage would be modified as needed by idioms.