The answer isn’t so easy. Many designers, museum directors, even visitors would think of a push button as interactive.
I would differ. The meaning of Inter – “occurring between“, a push button could be thought of as interactive but the communication between the participants (the “button and the visitor” is very limited with only one outcome). My definition of an interactive exhibit requires “open ended” or “branching outcomes”. A push button only has one outcome with out “branching outcomes”.
I would define a sand box as an interactive, there are almost unlimited outcomes between the “participants”. The access to the exhibit is immediate, a sand box requires very little explanation for interaction you can just start using the sand.
The best interactive exhibits have very limited action on the part of the exhibit, the sand is doing very little until you start to use it, then the action can be with the sand, other participants (past, present and future) as well as the conversation with other participants that accompanies such interaction, often the most important part of the exhibit dialogue is with other participants.