This Jacques was not born or baptised in Leiden.
The facts are :
He was married to Lea Villeman 1656 in Frankenthal.
They had 3 children there.
In 1659 they were in Leiden, in april 1660 they left. Probably for Frankenthal again because a child was born there in 1661.
In 1662 they went on a boat to America, with two childeren, (so one child passed away)
There is (still) nog evidence that he was from Amsterdam or from Leiden. No mention whatsover for Rachel Gelton.