Are the first and second divorces, which occurred within a short time frame and for the same mistake, considered one divorce or two divorces? And is it permissible for the husband to take back his wife after the third divorce based on this consideration?

If the situation is as you described, you have issued three divorces to your wife, and she has become irrevocably separated from you, making her unlawful for you until she marries another husband in a sincere marriage and he consummates it, and then he divorces her; 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 [this latter husband] divorces her, there is no blame upon them [the former husband and wife] if they return to each other and feel that they can uphold the limits of Allah. These are the limits of Allah, which He makes clear to a people who know." The fact that the second divorce occurred under the same circumstances or shortly after does not prevent its occurrence, especially if it was after reconciliation. One must be cautious about hastening to divorce due to its serious consequences.