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Mark already said that only the highest source of chill with this effect will take precedence. So you use it on 1 skill as kind of a utility to give extra damage to other cold skills and itself.
Now for the gem itself. There are 4 sections to it, but only 2 will ever apply to any one part of incoming damage. Chilled from hits. Chilled from ground effects. If chilled by both a hit and a ground effect, only the higher chill takes precedence. Both these groups add a bonus to on hit damage taken, and damage over time taken.
The last confusing bit is that "increased damage taken by chill effectiveness". This only apply to damage by hits. Damage by dots always get 29%. Chill effectiveness is how much the chill slows down the target. This will never get above 30%. So the damage bonus for hits can go up to 30%, but will mostly be lower. Especially if you don't invest in chill effect.
The last confusing bit is that "increased damage taken by chill effectiveness". This only apply to damage by hits.
No it doesn't. It applies to cold damage, like it says - if it was restricted to damage from hits, it would say so, just like the other line specifies it's only for damage over time. Since cold damage over time is inherently cold damage, both those modifiers will apply to cold dot.