Sorry for a slow response. The reasons I suspect the Leicestershire Carter family as that of Thomas of Lancaster VA because. 1. bap. date of 1630/31 would make that person "about 70 years old" in 1700 when he died and a birth date of Oct. 1630 for this Thomas is the right age; 2. Family names are the same for both families at least for children mentioned; 3; Thomas of VA named his first son Edward often the custom to honor the grandfather with the same name; 4. Thomas of VA was among the gentry class and Edw. was styled a Gent.; 5. When Edw. died in 1673 again called a gentleman, he likely would have had a will possibly naming Thomas of VA [I have not found a will]; 6. Thomas of VA's inlaws were the Dale and Skipwith families that lived in Leicestershire.
So, as you can see, lots of circumstantial evidence for his parentage but so far no solid facts to prove it. I am looking for sources in England to prove this parentage and welcome suggestions.