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I would suggest that significant weight be given to 1603 as being the best guess and approximate DOB for Abraham Perkins based on the following sources:
MORTENSEN, Paula Perkins, English Origin of Six Early Colonist by the Name of Prekins (363 So. Park Victoria Drive, Milpitas, CA 95053-5708, Paula Perkins Mortensen, 1998), Higginson Book Company, 148 Washington St, Salem MA 01970, page 08-09.https://dcms.lds.org/delivery/DeliveryManagerServlet?dps_pid=IE76427.
NOYES, Sibyl, LIBBY, Charles T., DAVIS Walter G., Genealogical Dictionary of Maine and New Hampshire ([database online] Portland, Maine, USA, The Southward Press, 1928), Ancestry.com, Surnames, O-P; Page 555 of 809 [541]. Record for Abraham Perkins https://www.ancestry.com/interactive/48566/GenDictMENH-008636-706#?imageId=GenDictMENH-008471-541
TORRY, Clarence A., New England Marriages to 1700 (Online database. AmericanAncestors.org, , 2008), AmericanAncestors.org, Volume 2, page 1165 of 1482. Record for Abraham Perkins. https://www.americanancestors.org/DB1568/i/21175/1165/426899281.
TORRY also makes references to the following sources: {Reg. 10:215, 12:79, 39:81, 50:34; Hampton NH 908; GDMNH 541, 773; Salisbury Fam. 282; Conant 156; Pillsbury And. 339; EIHC 13:92; Warner-Harrington 789; LBDF&P (1913)25}
All of these sources favor 1603 as his DOB, making Alice [1st] his mother.
The images in the Warwickshire, England, Baptisms, Marriages, and Burials, 1535-1812 records are not only difficult to read, but partial information can be misleading. There is an Abigail Perkins dau of Isaac bp about 8 Nov 1607 if I am reading it correctly and an Abraham bp in Jan 1608 (but you cannot read a last name for Abraham or his parents name). Also, these entries are on consecutive sequential pages so the birth/baptism events are a problem because they are only a couple of months apart.
I believe the better argument is that Abraham was the last child born, in 1603, to the 1st Alice who may have died as a result of this childbirth on 17 Jun 1603.