Here's the Bartons in my line (Sorry no connection to the Maxsons or to any of the other families you mentioned.)
-- Hepzibah Barton, b. 1794, Whittingham, Vt.; m. Sylvester Satterlee Jr. (a Rhode Island native), abt. 1810, NY.; d. 1863, Livingston Co., NY.
-- Jonathan Barton, b. 1754, Spencer, Mass.; m. Hannah Dix, 1781, Whittingham, Vt.; d. 1830, Bolton, N.Y.
-- Timothy Barton, b. 1732, Leicester, Mass.; m. Hepzibah Stowe, 1753, Southborough, Mass.; d. 1796, Chesterfield, Mass.
-- Joshua Barton, b. 1697, Middlesex Co., Mass.; m. Ann ???, about 1720, Mass.; d. 1773 Spencer, Mass.
-- Samuel Barton b. 1664, Salem Village, Essex, Co., Mass.; m. Hannah Bridges, abt. 1690, Salem Village; Mass.; d. 1732, Oxford, Mass. (Samuel is important because of his connection with the Witch Trials in 1692. He was a witness for the defense of Elizabeth Proctor. His mother-in-law Sarah Towne Bridges Cloyse was acquitted of witchcraft after her sister Rebecca Towne Nurse was hung. Samuel and his family, including his mother-in-law, left Salem Village the next year and were instrumental in founding the town of Framingham, Mass.)
-- Matthew Barton, b. abt. 1639, Cape Porpoise, Mass.; m. Martha ??? (2nd wife), abt. 1663; d. ???
-- Edward Barton, b. 1608-1618, Eng.; m. Elizabeth ???, 1639, Marblehead, Mass.; d. 1671, Cape Porpoise, Mass. (now Maine)
-- Marmaduke Barton, b. abt 1590, Eng.; d. bef. 1696, England
Even if your Bartons are not clearly in this list, I think there's a good chance they are related. Do you know anything about their religion? That usually played an important part in settlement in Rhode Island.
-- Frank Taylor