These terms are all subject to interpretation as there is no standard as far as I know. I experience different definitions per term from team to team, even within the same organization, let alone across a region, country or even international.
In my own team, we only have first and second tempo, as we define it as the moment the ball is at 'hitting height'. First tempo is before the ball is at its highest point, still ascending. Second tempo is when the ball is already descending after reaching its highest point. In this system, a 0th or 3th tempo does not exist, although we comically refer to them with some clumsy, bad sets (e.g. a set that does not reach over the net)
Also note that the tempo of a set does not define the whole set. It is just used to describe a certain aspect of it, so a lot more context is needed to describe the exact set you want to give/receive. For example, a back-row attacker could ask for a first tempo set at the middle of the court. Or the other way around, if the midplayer asks for a set behind the setter, the setter on his turn would probably ask 'first or second tempo?'