Is one who swore not to tell anyone something, then let slip and told part of it, considered to have broken their oath? What is the ruling on swearing based on a strong presumption, such as saying: "By Allah, they will not do it"? What is the ruling on doubting whether one swore not to greet a specific person, and are there any consequences for that?

The question indicates the presence of obsessive thoughts (waswasah), and their cure is to ignore them. Nothing is required of the questioner in the two mentioned cases, because doubt concerning an oath does not necessitate expiation, and the default is that no oath was sworn. Swearing an oath based on a predominant assumption which later turns out to be false, and an unintentional oath that slips from the tongue, are both considered by many scholars to be "laghw al-yameen" (vain or idle oaths).