How do certain actions, whose doers do not intend thereby to seek closeness to God—such as shaking hands or saying "May God accept [your deeds]" after prayer—align with the definition of bid'ah (innovation) as "a way in religion that parallels the divinely prescribed way, and through which closeness to God is intended"? And is it sufficient for an action to lack a basis in Islamic law for it to be an innovation, or is the intention of seeking closeness to God also required for it to be so?

The definition of innovation (Bid‘ah) is that which is intended as an act of drawing closer to Allah, for which there is no evidence in the Sacred Law. Thus, consistently shaking hands and supplicating after prayer, saying "Our Lord, to You be all praise and gratitude" (with the addition of "gratitude"), and celebrating the Prophet's birthday are considered innovations; because these are actions intended to draw closer to Allah, and they were not mentioned in the Sacred Law. As for that which falls under the category of customs and is not intended as an act of worship, it is not considered an innovation.