Chesapeake, Virginia, is a large city, not merely a "town," with a population of about 225,000. It borders North Carolina, but not Delaware.
None of the rewritten sentences have enough articles.
Chesapeake Bay is sometimes called "the Chesapeake" (with an article), but it's not a "region," it's a body of water. Geographers call it an "estuary." Anyone who lived in the Chesapeake would have to be a fish or other aquatic creature. If one live on the Chesapeake, one could live on a boat or live someone along the very long shore of the bay. The bay doesn't border on the state of Delaware, either.
Now, the Chesapeake region is a possible set of words that would describe the land all around the bay. I don't think it has an agreed definition. In discussions of American colonial history, Virginia and Maryland are sometimes referred to as the "Chesapeake colonies." I can imagine an article in a journal discussing some common institution in those two colonies titled, say, "Governors' Councils in the Chesapeake Region, 1700–1775." One might include the colony or state of Delaware in the Chesapeake region, but Delaware in the colonial period was an offshoot of Pennsylvania, and it borders on a bay of its own, Delaware Bay. A region consisting of the colonies or states of Maryland and Virginia would border on the colony or state of Delaware, since Maryland shares most of Delware's land border.
New York, Pennsylvania, and New Jersey are commonly called the "Mid-Atlantic" states, and are called by historians the "Mid-Atlantic" colonies.
And that might be more than I can get away with.