A beginning is a type of origin, being a sort of temporal or sequential origin, the initial moment of something. On the other hand, an origin generally is the cause or source of something (hence the 'derived' part of the definition you provided). Other definitions of 'origin' include "rise, beginning, or derivation from a source", "something that creates, causes, or gives rise to another", and "the beginning or cause of something". So while 'beginning' is commonly recognised as an aspect of what 'origin' means, it is also commonly recognised as being a 'cause' or 'source' as well.
In this sense, you could argue that something can have an origin but not a beginning, or at the very least that the two aren't necessarily the same.
Let's take actuality, for example. All of actuality, "what is" has its origin in potentiality, "what might be", as before anything can actualise it must first have the potential to do so. In other words, in order for something to exist in actuality, it must first have existed in potentiality. This makes potentiality the source and origin of actuality. On the other hand, the beginning of something existing in actuality is the moment it transitions from potentiality to actuality.
