At best, you could suggest that the English version of the Bible got the translation wrong, and the story was meant to say that God created Man - as in the race of man - out of the dust, and woman - as in the race of woman - out of one side of Man. You could argue that by race, it means there was a whole race of human beings created in this manner. However, that argument kinda falls apart when Adam, and Eve have two sons - Cain, and Abel. Kinda hard for an entire race of people to have two, named, children.
So, by the time we get to Cain, and Abel, we are forced back to the realisation that the story of Adam and Eve is one of two things:
Either it is an allegory of the origin of Man - which is what most modern Christians, and, I think - although Ian would have to confirm that for me - most modern Jews, say,
Or it was a true, and accurate description, assuming that one accepts the Bible as a valid source, of the origin of humanity.
Your version of "adam" falls apart with the having children part.