2. "The Pilgrims worshipped at the Pieterskerk, which is the 'real' Pilgrim church."

AANSWER: The Pilgrims never worshipped in the Pieterskerk. The Pilgrims as a whole worshipped in English, and Winslow mentions that Robinson's house was large. We assume their services were held in Robinson's house until that became illegal in the summer of 1619. (In 1619, in a measure to suppress the Remonstrants (Arminians), all discussion of religion in private homes was forbidden! This has to have contributed to the unease inspiring the Pilgrims to look elsewhere.) Robinson's house was across the churchyard from the Pieterskerk. Robinson died in 1625, and like everyone living in the parish area of the Pieterskerk, he was buried in that church. That is why there is a memorial to him there.

When Winslow mentions communion with the Dutch, he explains it quite differently from attending any Leiden church. He says that Pilgrims who knew Dutch, when they visited LONDON, went to the Dutch church there (Austin Friars), and took communion with the Dutch, specifically naming John Jenney as a Pilgrim who with his family attended the Dutch church in London. By doing this they attended church without attending an Anglican church! Pilgrims who travelled in The Netherlands away from Leiden also attended Dutch churches without any offence, according to Winslow. (same reference as his comments on the Walloon churches).