I thought you might enjoy this discussion I found at BioLogos:
From the thread linked below:
“You are basically on the right track, though I don’t think that Adam is named in Genesis 4 (nor Genesis 2-3), despite the rendering of some translations. Here is footnote 12 from my published article, “From Primal Harmony to a Broken World: Distinguishing God’s Intent for Life from the Encroachment of Death in Genesis 2–3.” Chap. 7 in Earnest: Interdisciplinary Work Inspired by the Life and Teachings of B. T. Roberts , ed. by Andrew C. Koehl and David Basinger (Eugene, OR: Pickwick, 2017), 145–173.”
“There are four places in the narrative of Genesis 2–3 where ’ādām appears without the definite article, but none of these is a proper name. According to 2:5, “there was no-one [lit. no ’ādām] to till the ground.” In Gen 2:20, 3:37, and 3:21 we have lĕ’ādām (to/for the human); here the preposition lĕ (to or for) is appended to ’ādām) without the vowel change that usually indicates a definite article (lā’ādām).”
“However, in the first case (2:20), the same verse also uses ha’ādām (the human); and it should be remembered that there would have been no distinction in the original Hebrew consonantal text (so the vowel pointing that the Masoretes introduced, which we have in our current Hebrew Bibles, may be idiosyncratic).”
"Gen 4:25 is the first clear use of ’ādām without the definite article (“Adam knew his wife again”). Yet Gen 4:2, which first mentions the man knowing his wife, has ha’ādām. In Gen 5:1, which begins a genealogy, we finally have the proper name Adam clearly intended.