The short answer is that to define a paraconsistent formalism, you have to define decision-making rules. The rules must say what can be concluded from an inconsistent set of formulas. Simply put, your proof theory reflects the decision-making rules that you like, and which you built into the language when you made it.

Even if you are not making the paraconsistent formalism yourself, but, say, are working with a specialist, then you need to define informally, but clearly the decision-making rules first. They should ideally be specific to the problem domain, or problem class that you want to be solving with that formalism.

These are simple observations, and have a crucial implication: there is unlikely to be one best paraconsistent formalism, because there are no universal criteria for which conclusions are valid, when you have an inconsistent set of formulas.

So another important implication is that if you say “I am using paraconsistent formalism X, which someone else made” then you need to make it clear how the decision-making rules in that formalism are good for what you want to use that formalism for. Otherwise, you are picking one, among many, and remains unclear why that one is more relevant than another.

So when making a new, or picking an existing paraconsistent formalism, you need to make it clear why that formalism draws, or better, prefers some conclusions, rather than others, when it is given a set of inconsistent formulas.