What conditions characterise "cheap-talk" games
Your general question is in my opinion far more interesting than the course problem, so I will answer that.

What conditions characterise “cheap-talk” games?


As you have noted yourself, the talk needs to be cheap, meaning signals/messages should have no direct effect on the players' utility. Indirect effects are allowed. Even in cheap talk games it is possible that the beliefs of a player are updated by a cheap message, hence his response changes altering the outcome. Simple examples include coordination problems.
A second necessary condition is that all messages are available to all types. hence there are no type specific message spaces $M_i$, only a shared message space $M$ from which all types of the sender can freely choose their message. Without this, a signal, though costless, could carry information, as certain types are incapable of sending it.


It's a cheap-talk game, so what?

If these two conditions are met then the game is a cheap-talk game. Why is that important?

Let us call a version of the cheap-talk game where we leave out the signal round the simple variant. For any equilibrium of the simple variant there is a payoff equivalent equilibrium of the cheap-talk game. Similarly, for any 
equilibrium of the cheap-talk game there is a payoff equivalent equilibrium of the simple variant. Hence it seems that the ability to signal did not fundamentally alter the set of equilibrium outcomes. The reason for this is that the signal did not carry any information, as it was easily imitable by other types, it was 'cheap'.