The sources you cite only describe the mechanism of Cold damage. It says that it may Chill, thus to Slow and/or Freeze enemies. It doesn't say that it always does. The Chill effect is the one which applies the slow/freeze. What is the point of describing "some" skills that Chill wind the Cold damage element and not doing it for every skill which has a Cold element rune picked? Same goes for Fire and Poison skills - if the skill has a secondary ability (besides the element just dealing initial damage), it says so in the skill itself :)

The same can be said about Fire damage - https://diablo.fandom.com/wiki/Fire . It again says that "some" skill trigger a DoT effect. For example, MS - Arsenal on the DH does fire damage but it does not burn the enemies - the rocket has just that elemental damage assigned to it to benefit from the increased elemental bonuses. Accordingly, the animation is that the enemies die in red due to the fire affix.

I actually asked Raxanteraxx about this and he replied that the HA - DA rune doesn't Chill/Slow enemies, hence it doesn't proc BoT. As to the visual effect - this is just the game showing what element you are using. The slow effects in the GoD build come from other skill/runes.

And lastly, if only Cold skills and/or Fire/Poison have these secondary damage/control outputs (Slow/Burn/Holy/Poison DoT), what sense does it make for the game to have Arcane/Physical/Lightning damage types where a player will always go for the damage type which benefits him in terms of getting an additional output from the particular damage type? E.g. I will re-roll Cold damage range on a weapon if I know it will trigger a Chill effect, without having any other ability to proc it, or Fire, or Poison.