Dear Bernie:
We agree with your definition of the word 'dominion' or 'rule' as used in the context of Genesis 1, which also includes verse 28.
However, based on the way this word is used elsewhere in the Bible, it appears that the word actually has a more complicated meaning of ruling over God's creation in a way that will not only loving care for it, but that will also conquer all ungodliness, which is exactly what Adam failed to do when Eve was tempted by the serpent.
In the Love of the Lord,
Frank and Mary