Off the top of my head, something like being able to make threats while pursuing your natural play.
So for example lets say the opponent has a weak king and you maneuver to attack it.
Sometimes those maneuvers will create threats along the way. If so, then I'd call that initiative.
But other times, your building-up moves can be ignored, and the opponent e.g. spends his time preparing for counter play. In this case you wouldn't have an initiative.
---
I guess another way to say it is when you can force your opponent to respond to your moves for a protracted period of time. However in my mind "initiative" is more than that, because if you're playing poor moves just to force a response then it's sort of a fake initiative. That's why I include the criteria that it must also be pursuing some logical play. Attacking the king is one, but it could just as easily be attacking a pawn structure, or a sector of the board e.g. trying to infiltrate on the queenside.
What criteria(s) could be used to determine who has the initiative?
Your contributions are highly valued. Please feel free to explain yourself clearly. Thanks for inputs.