No, all of it is true. The institution of slavery is completely different from the one in the west. Ottoman Empire didn't have a plantation based economy, rather is was served to supply state servants and also household servants. Furthermore, all the highest administrators and soldiers were legally slaves of the Sultan. Thus, slavery in the Ottoman Empire cannot be compared with slavery in a plantation in the Americas.

By the way, Ottomans kept all the slave record in Istanbul. Who had slaves, when they acquired when they freed are all documented. The numbers are a few thousands if you exclude slaves of Sultan and other government officials (who were slave themselves by the law), it was about a few hundred thousand total.