What is the ruling on the divorce that occurs to the wife in the three aforementioned instances, and do the first and second divorces count, or do I owe an expiation?

The occurrence of divorce is a matter of disagreement among jurists. However, since you acted upon the ruling of one who decreed its occurrence, it is not permissible for you to revert to the opinion of one who decreed its non-occurrence.

Accordingly, your wife has become irrevocably divorced from you with a major divorce (baynunah kubra). She is not lawful for you until she marries another husband in a marriage of desire, and he consummates the marriage with her in a real consummation, and then he separates from her. This is based on the Almighty's saying: "And if he has divorced her [for the third time], then she is not lawful to him afterward until she marries another husband. And if he [second husband] divorces her, there is no blame upon them both if they return to each other, provided they think that they can maintain the limits of Allah. And these are the limits of Allah, which He makes clear to a people who know." (Al-Baqarah: 230).

One should be cautious of hastening into divorce, for its consequence is often regret.