In this case, you need to check more dictionaries; none of the options listed in your link really fit the way the word is used in that sentence.
A more comprehensive list is at Merriam-Webster's website, where it includes
3b: active and effective in social or economic function rather than merely maintaining peace and order
4a: indicating, relating to, or characterized by affirmation, addition, inclusion, or presence rather than negation, withholding, or absence
In your sentence, the intention is that the State does not simply sit around and wait for people to tell it what to do and what cases to bring; it goes out and actively seeks out instances where it needs to take action, open investigations, bring charges, prosecute wrongdoers, and so forth.
A common synonym for this sense of "acting positively" is "being proactive".