A volitional verb expresses an action that is controlled by the will of a sentient actor. It’s not really an intrinsic property of the verb, it also depends on context. The actor needs to be sentient, and the action as a whole must be controllable.
Note that wasabi isn’t actually talking about the verb at the end; for ために to express an aim (in order to …), the part in front of it should be volitional. Maybe that’s also helpful to differentiate: ために is used for aims (things you decide you want to do, which can be expressed with a volitional verb), while ように is used for ideal situations (situations that you would like to happen but can’t consciously create, and would therefore be described with a non-volitional verb).
In the ように section at the bottom there are some good examples that try to clarify the difference, e.g.:
雨が降るように（祈った / 祈りました
雨が降るために（祈った / 祈りました
Here, ために can’t be used because the action (rain falling) is not volitional, and it’s also not the same subject (it’s not the rain who is praying).
ケンに会うように日本へ（来た / 来ました
ケンに会うために日本へ（来た / 来ました）
Here, the action (meeting Ken) is volitional, and it’s the same subject, so ために should be used.
I didn’t think about it before, but the ために grammar point has an example sentence that’s sketchy in this regard: In “日本語が上手になるために”, the action is not volitional, and it’s also not the same subject before and after ために. This should be either ように (it is an ideal situation), or an active, volitional construction such as 日本語を上手に話すために (in order to speak Japanese proficiently) or 日本語を上達させるために (in order to improve my Japanese) or something.
