Here in the final chapter of this book, the SAB has only one question: "How did Saul die?" The page pointing out the apparent inconsistency gives four possible answers. Now while one can easily dismiss the account of the Amalekite as a lie (after all, he had no witnesses, and apparently assumed that telling David that he had killed Saul would curry favor) I don't think it's necessary, and besides, preachers who cover that story rather like to point out the irony of Saul being killed by an Amalekite, since he was ordered by God to wipe them out and failed.
The thing is, all four of these explanations can be correct in a sense. As I've said before, in some sense, everyone gets killed by God, so no need to explain that. Saul was killed by the Philistines because he died while in battle with them, being struck a mortal blow by Philistine arrows. Knowing that he was as good as dead he endeavored to kill himself rather than be subjected to torture. It may be that he failed to kill himself, and the Amalekite ended up landing the final death blow upon Saul.
The battle is a resounding defeat, and Saul dies along with his three sons, and many of the troops. The next day, Saul's body is captured by the Philisties, who hang his head on the wall of the city of Bethshan.