The choice of prepositions like "from" and "by" depends on the kind of relationship being expressed between the verb and the source of information. In "understand this from their actions", the preposition "from" shows the origin or source of the understanding. You're getting the idea or insight based on what they did — their actions are the evidence that leads to understanding. In "determine by her gait", the preposition "by" shows the means or method used to determine something. Here, her gait (the way she walks) is the way or method through which the judgment is made. So while both prepositions relate to cause or reasoning, "from" tends to show where the idea comes from, and "by" shows how the idea was figured out.

The logic is: "from" goes well with "understand" to emphasise that the "actions" are a direct source of insight; while "by" goes well with "determine" to emphasise that the "gait" is merely an instrument used to help - not a complete signal in itself. "From" is good for referring to one end of a simple journey. EG: I went from London to Kiev. "By" is good for referring to something that helps or happens along the way. EG: I went from London to Kiev, by Plane. And by the time I got half way, I was tired. That's the academic logic, and that's half of the truth. The rest of the reality is, people speak like that because they are copying other people before them. If people used different words, people would copy those different words out of habitual tradition. This is the realism neglected by the common academic explanation.