Oo-er.
The Visitation book is from a manuscript collected by Peter le Neve, apparently a copy of the 1634 visitation, with many additions by later hands, printed in italics. No italics here, so Matthew had 3 sons married by 1634 and the eldest had a kid. The date against his name would mean that he was the informant, so obviously living.
Our Matthew has 8 kids with the same names, but dates a century earlier.
So he's a Conflation. The Matthew with the descendants isn't the same Matthew that the wife and the other kids belong to. The descendants must have dates that fit lower down. The wife and the extra kids have been given made-up dates to fit.
Elizabeth Bowchiew's been mentioned before.
(And now I'm thinking Richard Bowle d 1626 actually is Elizabeth's father)