So, as the question says by itself: what's the context when I should use the word mist and the right context for fog? And haze?
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.
There is an overlap between the three.
Both fog and mist apply to conditions in which water is suspended in the air, which decreases visibility.
Part of the reason for their being two such words is that once there was mist (goes all the way back to Old English) and foggy (coming from the separate sense fog which is a type of grass, or metaphorically a mess), and then fog came from people shortening foggy.
It is generally taken that fog is thicker than mist. Just where the difference is between the two depends on who you are asking. In the UK, as far as driving laws go (when it's required and when it's forbidden to use fog-lights) fog is when your visibility is 100m or less, but other people who have a reason to set a precise rule on the matter may set it differently. (Having started to learn to drive under the UK rules, and having more lessons under the Irish rules, I never heard anything more precise for the latter than "that sounds about right" after I explained the British).
Haze was historically used to mean a particularly thick fog, but is now taken to mean a particularly light case. As well as covering the case of a mist/fog that is very light, it will also cover some other causes of reduced visibility, such as the distortion caused by refraction in particularly hot conditions ("heat haze" is used to refer to this specifically).
In figurative use, they overlap similarly, and similarly fog is generally heavier and haze lighter: A couple of beers might have you a bit hazy, while a night's drinking followed by a couple of joints would have you decidedly misty.
If you need a precise answer, then it depends on the context you need to be precise in relation to (e.g. the 100m rule for driving in the UK). In everyday usage, the difference between them is much looser.
Then the weather guessers had to mess things up. Up until the `80's Fog was <3mi visibility, mist was between 3 and 7 miles visibility. and haze was >7mi but not unlimited. Then everyone went to the METAR system. Now mist is both fog w/ visibility <7 miles AND drizzle with visibility <7 mi. In my mind they are NOT the same Drizzle gets everything wet, fog unless it is really thick, doesn't.