lilbra on Family Tree Circles

A distant cousin wrote back in 1931; When I was last in New York City,the librarian of the finest city library I have ever visited, showed me a Pedigree of the Adams family going back to 1310. Our Adams line is notable in that we have already given two Presidents to the United States of America.
I am trying to trace the connection to which he refers but so far have been unable to trace a common ancestor, but have found , I think, the Pedigree mentioned which is as follows;
SIR JOHN AP ADAM KT. LORD AP ADAM m ELIZABETH daughter of SIR JOHN GOURNEY of BEVERSTONE and TODDENHAM GLOS.Several generations resided at CHARLTON GLOS
SIR THOMAS AP ADAM AND OTHERS
SIR WILLIAM AP ADAM
SIR JOHN AP ADAM
THOMAS AP ADAM

SIR JOHN AP ADAM KT m MILLICENT daughter of SIR MATT BISYLLS

SIR JOHN AP ADAM alias ADAMS m CLARA daughter of MR ROGER POWELL

ROGER ADAMS m JANE daughter of MR ELLYOTT
THOMAS ADAMS m MARIE daughter of MR UPTON
JOHN ADAMS m JANE daughter of MR RENNELEGH
JOHN m CATHERINE daughter of MR STEBBING
NICHOLAS married NO ISSUE
JOHN m MARGERY daughter of MR SQUIER (SQUIRE)
GEORGE HAD ISSUE
RICHARD (son of JOHN) m MARGARET daughter of MR ARMAGER

ROBERT ADAMS m ELIZABETH SHARLON
GEORGE
HENRY
AMBROSE
JOHN

Here I become confused as according to the LDS the parents of GEORGE HENRY AMBROSE and JOHN were WILLIAM ADAMS b 1555 in TRENT SOMERSET and a MISS DORRINGTONb 1557 in DEVON ENGLAND,and HENRY born 1583 was the Gt Gt grandfather of JOHN QUINCY ADAMS.
Could it be a coincidence that two Adams' families gave their sons the same names?