Regarding the LNAB, I refer you folks to the New Netherland project naming convention for LNABs (see the project page), wherein we try to use the name found on the earliest record (ideally a church record) that gives the person a name that is clearly their own name (not, for example, their father's patronymic name or a patronymic name devised from the father's first name), with the spelling found on the record. Before this admittedly arbitrary policy was adopted, we had lively discussions (call them arguments if you will) about topics like (1) the "correct" LNAB for a child who was presumed to have had a patronymic name at birth, but there is no record of the person being called by a patronymic name (for example, we wondered that if the father was recorded with first name of "Thijs," should his LNAB be the patronymic "Thijsz" or a variant like "Tyszen" or "Matthyssen" or myriad other possibilities, none of which ever appear in the person's records?) and (2) competing theories on the national origins of immigrants who might have been French, Dutch, English, Frisian, German, or Scandinavian -- and whose LNABs were proposed to be based on members' personal opinions regarding the nationality.
Instead of imputing a last name based on supposition and speculation, we use the name on the record as the LNAB. All other last names go into the Current Last Name or Other Last Names data field, and all variants (and any competing theories of national origin) should be identified/discussed in text.
For this man, we seem to have evidence for his place of origin (Rijnsaterwoud) -- but I don't remember where I saw mention of a record that showed that, plus a few records from that place (the land records that Remko found) that are likely to be his. It looks like some of the "records" cited for him are not faithful renderings of the records, but rather were reinterpreted in English. If the records really do refer to "Gerrit" or "Gerret" (given name) and "van Sweringen," then one of those given names and the last name "van Sweringen" seem like good approximations of his given name and LNAB.
NOTE: The names "Garrett" and "Van Swearingen" look to me like English translations of Dutch names. And "van" is a Dutch-language preposition that should be rendered in lower case.