ために is used with verbs that indicate volition in order to express an intention to produce a desired result. ように does not use verbs of volition and simply states a condition that is a desired result without implying direct fulfillment of those conditions through a specific exerted effort.
There's an explanation here that explains it pretty well, so I'll translate some of what it says:
First they give some example sentences:
Examples a and b both use verbs of volition as in "in order to protect nature" or "in order to travel around the world." Examples c and d, on the other hand, use verbs of possibility or intransitive verbs. It's stating a condition rather than an intention as the goal. Note that in d the change in subject does not affect the 'intentionality' of the verbs.
When creating these sentences we have to think about what the attitude is toward the end result. For example in b the kind of thought process is "I want to travel around the world. Therefore I saved 500 man yen." Compare this to example d where the thought process is more like "It's nice if the whole class understands what the teacher is saying. Therefore the teacher spoke slowly."
So it's about intentional actions versus desired conditions.
Now to offer my own two cents to this, your original instinct about action verbs I think isn't wrong. I'm not sure if "action verb" and the Japanese term 意志動詞 overlap perfectly, but I'm assuming that most volitional verbs are indeed action verbs. It's more important to think about the distinction between conditions and volition, though.
So in your examples, the first one expresses that the condition brought about by not being late for your plane is a good one, and you are taking that action as a step toward fulfillment of that condition. "乗り遅れる" would not normally be considered a volitional verb because the lateness kind of implies something that you don't want. It's a negative verb that is basically another way of saying like, 間に合って指定された時間に乗れる, which ultimately is a condition. If you just use 乗る then things are different. In the second, "answer a question" is not a condition. It is an action that you are doing out of your own volition, a desire to answer the question, and you are raising your hand as a direct means to achieve that.