Piezo speaker with distorted sound
A likely cause for the occasional sound distortion is that the CR2032 cell is unable to sustain the current draw of the piezo buzzer plus the ATTiny for extended periods. This would be likely if:


The buzzer is operated several times in quick succession, not giving the cell a chance to recover
The ATTiny itself is consuming more power at certain times due to parts of its code


This hypothesis can be validated by running the buzzer directly off a partly depleted CR2032 for a few minutes - If the buzzer sound distorts like in the circuit after a bit, then the culprit has been identified.

Possible Solutions:


Use two or three CR2032 cells in parallel for greater current sourcing capacity
Use a reservoir capacitor of a relatively large value, to tide over the high load periods - This won't solve the problem, just alleviate it a bit
Use a battery with more current delivery capacity