I agree that the naming pattern is strong evidence in favor of the Poortmans. There also is a primary source for Jan Poortman's baptism and parents.
The Putmans were late 16th and early 17th century pastors and lawyers in the Twenthe region of the province Overijssel.
A book on early Twenthe pastors was written by historian professor Paul H.A.M. Abels in 1984: "De broederen van Twenthe; een studie van de eerste Twentse dominees (1597-1678)" ("The brethren of Twenthe; a study on the first Twenthe pastors (1597-1678)").
A private tree with about 130,000 pre-1812 persons has a branch of Putmans starting with the 16th century Jan. These Putman persons have citations of that book as source.
Church records of the relevant villages Goor and Delden are preserved starting second half of 17th century, so after the supposed 1644-1645 birth of John (probably meaning Jan or Johannes) Putman.
Anyway, the tree has no record for such Jan as son of Rutger, but this Rutger did have children baptized Sep 1644 and March 1646. Another child in this 18 month gap is possible in theory, but quite unlikely I think.
So that leaves the col Putman scenario as possibility. No traces of this branch, starting with the 16th century Jan's brother Abraham, I could find. But why would this mysterious Abraham that studied law have to flee for religious reasons when all of the family did not, being of the prevailing Dutch Reformed relgion?
I also tried to find the Dutch version of the "History of the Putman Family in The Netherlands", but could not find any reference to a likely Dutch title or Adolf/Adolph Hendrik/Hendrick Putman as author of any book.
I agree that Jan Poortman is the most probable progenitor of the New Netherland Pootman/Potman/Pottman/Putman branch.
Given the Dutch Putman family tree (reliably sourced in my opinion) it would not be too hard to add that to WikiTree.