What is the difference between a covenant (ʿahd) and a promise (waʿd), knowing that the expiation for a covenant and a vow (nadhr)—if fulfilling them becomes difficult—is like that for an oath (yamīn), whereas a promise is not like an oath in terms of expiation?

A covenant (ʿahd) is the preservation and care of something. It is also applied to a binding document that necessitates observance, and to entrusting something and placing it under one's responsibility. As for a promise (waʿd), it is the announcement of an action to be performed in the future that relates to another. They differ in that a covenant is a promise coupled with a condition.