Is a husband's divorce of his wife with the word "talik" or "taliq" without witnesses, but with the intention of divorce, and then taking her back without witnesses, considered a first and second legal divorce? Is it permissible to take her back after the third divorce in the same manner and before the 'iddah (waiting period) ends?

Your utterance of the phrase "Taliq" or "Talik" with the intention of divorce effects a divorce. Since you took your wife back after two previous divorces, your divorce of her after her apostasy and her return to Islam during her iddah (definition) is considered valid according to the Shafi'i school and one narration in the Hanbali school, and this is the more preponderant opinion. As for the Maliki school, divorce occurs simply by apostasy. Witnessing is not a condition for divorce or reconciliation, but it is recommended. With the occurrence of the third divorce, your wife has become unlawful for you until she marries another husband in a marriage of desire and he consummates the marriage with her and then divorces her. It is recommended to consult a Sharia (definition) court.