I am not an expert on US genealogy, but I did a little digging around to try and clarify the problem to help you.
When looking into things, there were several things I noticed:
1. Your research notes on the profile of both the 1767 Elijah Peck and on William Peck are very clear. I think it's the right way to deal with the situation.
2. When looking at the sources for Elijah Peck, there is only 1 original source from the time period mentioned in your research notes. '"Connecticut Births and Christenings, 1649-1906", database, FamilySearch (https://familysearch.org/ark:/61903/1:1:F74D-D8N : 7 January 2020), Elijah Peck, 1769.", however, when following the link it shows an index. Definitely check out the original record there.
3. Some records, like the book History of the Abington Baptist Association, From 1807-1857, says Elijah Peck, born May 3d, 1767 in Warren, Litchfield county, Connecticut. While the index from the source in 2 says Elijah Peck, born May 3d, 1767 in Kent, Litchfield county, Connecticut. This is not just a case of two men with a different birth year, but also a different birth place. That sends up all sorts of flags for me, and makes it more uncertain they are the same Elijah Peck.
4. For William Peck, you do not have an original record for his birth. A good first step would be to try and find William in as many original records as you can, to see if you can glean more information about his parents that way.
5. Search both places (Warren and Kent) for original records pertaining to Elijah Peck, born in the 1790-1800 range. Track all candidates - try to see if you can eliminate them.
6. If that doesn't help, go with cluster research. Research William's siblings to hopefully find more information on the parents. Also, if the book mentions siblings of Elijah Peck, research them as well to see if you can link them with William or his siblings in a familial relationship.
This is not going to be an easy thing to resolve, unfortunately, It's not just going to be one birth record and done, because aside from finding and Elijah born in 1767 or 1769, you also need to prove it's the right Elijah. And maybe even that there is no Elijah in the other birth year/place, and that if there is another Elijah, that it's the wrong one.
I wish I could be of more help with actually pointing you to records, but I just don't know enough about US genealogy to help with that.
Good luck with this puzzle!