It seems that all your definitions collude with the same single definition - "as a consequence / result of"

Let's see how this fits into each of your quoted verses:

 1. Don't eat chametz. As a result of that, you will remember...
 2. The result of someone saying "I will have peace" is that he will be satisfied (i.e. in this verse it is a not a result of not observing the mitzvoth, but a result of his saying that he will have peace, i.e., this is his thinking.)
 3. The first לְמַעַן seems superfluous because the same word appears at the end of the verse. Allow me some time to research this further, and I'll try to edit an answer if I locate one. Nonetheless, the definition would still seem to mean "as a result of".

 3A. The 2nd לְמַעַן means - As a result of teaching his children... G-d will bring unto Avraham what he has told him.