Western Christians often had some odd ideas about Islam many of them contradictory. However what recently popped into my mind was well what did the Muslims think about their faith? Due to the nature of Islam and the fact that Muslim states had many Christian subjects they would have better general idea about what Christianity was. However did they understand or care about the split between the Pope and the Patriarch of Constantinople? Was there any Latin practice that really stood out for them? Was this something they would have understood as being analogous to their own splits i.e Shia vs Sunni?
Much more can always be said about this very complex topic! But I answered a similar question the past that might be helpful:
What stereotypes or preconceptions did the Arab world hold about Europeans during the Medieval era?