 A central approach to connexive logic is given by many-valued and model-theoretic semantics in terms of truth values or support of truth and support of falsity conditions. As explained in Omori and Wansing 2019, the semantics of several connexive logics can be described as either (i) modifying some standard (support of) truth conditions of conditionals of a certain kind or keeping standard truth conditions in combination with more complex model structures, or (ii) as tweaking the standard (support of) falsity conditions of certain familiar implications. Given the multitude of different connexive logics and the flexibility of the adjustment of falsity conditions in combination with standard (support of) truth conditions, this classification provides a general perspective.