Are the three matters—hastening the supplication's answer in this world, or warding off an evil, or storing it for the Hereafter—considered part of the supplication's answer, or is the supplication's answer limited to the fulfillment of what was requested in this world?

The Ahadith indicate that the supplication of a Muslim is answered, and that a servant will not be deprived of good from their supplication. Either their supplication will be granted promptly, or it will be stored for them in the Hereafter, or an equivalent amount of evil will be averted from them. All of this is part of the answering of supplication. Therefore, the servant should strive in supplication, having a good opinion of their Lord, content with what Allah decrees, and should not hasten the answer by saying, "I supplicated to my Lord, but He did not answer me."