They changed a couple of details but the broad strokes are correct
...as are the armor, banners, etc.
the extrended Directors cut makes this from an average movie to a very very good movie
Everyone, including William Monahan, the writer, knows that it is historical FICTION. They weren't trying to have a historically faithful retelling of the crusades. Just an exciting fictional story based around some historical events and people.
For the most part it had Europeans playing Europeans, and middle easterners playing middle easterners. I didn't see many white people playing middle eastern characters.
In some cases he even insists that the fiction shown in the film (e.g. the notion that Baldwin V had leprosy and that Sibylla killed him) is truer than what you will read in history books because their research uncovered facts leading to the conclusions they dramatised. Which is pure bunk.