If a husband conditions his wife's divorce on her betrayal of him, and he is unsure about defining his intention regarding the concept of betrayal—whether it is limited solely to sexual intercourse with another woman, or if no specific intention was defined—does the divorce occur only upon the act of sexual intercourse, or upon any act indicating betrayal, such as looking at other women or lying?

If you doubt the intention behind a conditional divorce oath, refer to the cause of the oath. If there is no known cause for the oath, it should be interpreted according to its common customary meaning. Thus, the oath is not violated except by an act of betrayal conventionally understood as such.

Ibn Qudamah said: "If the intention is absent, one should refer to the cause of the oath and what provoked it, as this stands in place of his intention due to its indication. If that is absent, his oath should be interpreted based on the apparent meaning of its words. If it has a jurisprudential customary meaning, such as prayer (Salah (definition)) and zakat (definition), his oath should be interpreted according to it and would encompass its valid form... And if it has no jurisprudential customary meaning, but has a customary meaning in common practice, such as narration (riwayah) or a traveling litter (za'inah), his oath should be interpreted according to it."

It appears that you are suffering from obsessive doubts (waswasah), so turn away from them and do not pay attention to them, for yielding to them leads to great evil. Seek help from Allah, do not give up, and occupy your time with what benefits you in your religious and worldly affairs.