What might explain why the Dutch believed that they had more of a claim on New Amsterdam than the British?

The Dutch believed that they had more of a claim on New Amsterdam than the British is:

They were the first one to claim the land.

Back when territories between countries still haven't been settled, the one that first settle the territory will have control over it.

it increase by 12% a year from 1812-1820

“i think it is very plausible that if there are five votes to end the death penalty, it could be with the people who are on the court right now,” - smith

so basically it is not clearly and consistently imposed