It appears snow forms in cold biomes where precipitation is set to false.
I noticed this when I created a custom biome where the temperature was really low and precipitation was false.
There is no visual snow falling, but the snow layers do form on the ground (see pics)

Custom biome:

{
  "temperature": 0.05,
  "downfall": 0.4,
  "has_precipitation": false,
  "effects": {
    "sky_color": 16753935,
    "fog_color": 16728403,
    "water_color": 65426,
    "water_fog_color": 329011,
    "grass_color": 16749055,
    "foliage_color": 15527168
  },
  "spawners": {},
  "spawn_costs": {},
  "carvers": {},
  "features": []
}

Something I noticed:
In the code under ServerLevel#tickPrecipitation, when deciding if snow should form on the ground, it uses the method Biome#shouldSnow, problem is this method just checks temperature to decide if it should snow, and never actually checks the precipitation value.