To be clear, I (parent AC) wasn't saying that the probability distribution is the wave function, just that it is given by it (which you confirm, it is the square of the amplitude). Now consider you make an observation and collapse the system to a single state. This state had a certain probability of occurring (again, given by the wave function). If you try to measure again, you will get the same state.

Only if you don't observe any orthogonal characteristics in the meantime. Consider a two-state system, with eigenstates |a> and |b> (for example, z-spin). Now consider an orthogonal basis |1> and |2> (for example, x-spin) which spans the same Hilbert space, such that

|1> = 1/\srqrt{2} |a> + 1/\sqrt{2} |b>

|2> = 1/\sqrt{2} |a> - 1/\sqrt{2} |b>

Now, suppose we observe the system to be in state |a>. Then if we perform an observation in the orthogonal basis, we will have a 50% probability to be in state |1> and 50% in state |2>. Suppose it's in state |2>. Now if we observe the first basis again, it's not in state |a> with certainty any more, despite the fact that we just measured it. It has a 50% chance to be in |a> and 50% to be in |b>.

There is no necessity to "restore coherence": the system is fully coherent throughout. This behavior does not happen with coins.