I think you are correct Destrudowoman. Let me start with a quote from Jung that provides a clue to how our mind works:
There are, moreover, unconscious aspects of our perception of reality. The first is the fact that even when our senses react to real phenomena, sights, and sounds, they are somehow translated from the realm of reality into that of the mind. Within the mind they become psychic events, whose ultimate nature is unknowable (for the psyche cannot know its own psychical substance). Thus every experience contains an indefinite number of unknown factors, not to speak of the fact that every concrete object is always unknown in certain respects, because we cannot know the ultimate nature of matter itself.
The underlined portion is the critical one, and "translated" is the key term. And this fits right in with what you write
archetypes are a product of the collective unconscious.
And the key word here is "product." It seems to me that the collective unconscious does not "contain" archetypes, rather, as you write it "produces" them. My speculation is that the collective unconscious is a universal human biological process that recognizes patterns. When the collective unconscious detects a pattern it "produces" an image. This is the "translation" that Jung writes about in the quote above. And, further, as you write it is the cultural environment we are born into that "names" the archetypal image.