No, nullify prevents active skills from being activated by other sources.

For instance, if a champ has a passive that says  "activates [a2] when [whatever condition] happens" nullify would prevent it. Some partner champs have a skill that activates when the partner does somethng, nullify would prevent that.

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Does the description stated prevent active skill from activated by "other source" ??

No!! Not at all...what it mentioned is under nullify debuff cannot have skill activated! Logically should mean active skill shouldn't be activated at all!! 

This is such a poor design or statement to mislead others!! Worst skillset ever!!

No argument that it's badly worded. It's been a trend recently to have odd skills with big wall of text descriptions that do a piss-poor job of actually explaining it.