Answer: Ibn al-Athir's account
Step-by-step explanation: I've never had a question like this in my life, but it would make sense for the first account because it's objective and straightforward whereas Fulcher's account has much more emotion in it. Like journalism, reports are aimed to be straightforward and unbiased as possible. This includes lacking the writer's own personal feelings. If we analyze by these means, it would make sense for Athir's accont to be more reliable.
Ibn al-Athir's Account:The population was put to the sword by the Franks, who pillaged the area for a week. A band of Muslims barricaded themselves into the Oratory of Savid [Tower of David] and fought on for several days. They were granted their lives in return for surrendering. The Franks honored their word, and the group left by night for Ascalon [a city on the coast]. In the Masjid al-Aqsa [the Temple of Solomon] the Franks slaughtered more than 70,000 people, among them a large number of Imams and Muslim scholars, devout and ascetic men who had left their homelands to live lives of pious seclusion in the Holy Place.
Fulcher's Account:Some Saracens, Arabs, and Ethiopians took refuge in the Tower of David, others fled to the temples of the Lord and of Solomon. A great fight took place in the court and porch of the temples, where they were unable to escape from our gladiators. Many fled to the roof of the Temple of Solomon, and were shot with arrows, so that they fell to the ground dead. In this temple almost ten thousand were killed. Indeed, if you had been there you would have seen our feet colored to our ankles with the blood of the slain. But what more shall I relate? None of them were left alive; neither women nor children were spared.