Here is the Wikipedia version:
Henry de Bohun, nephew of the Earl of Hereford, was riding ahead of his companions when he caught sight of the Scottish king. De Bohun lowered his lance and began a charge that carried him to lasting fame. King Robert was mounted on a small palfrey and armed only with a battle-axe. He had no armour on. As de Bohun's great war-horse thundered towards him, he stood his ground, watched with mounting anxiety by his own army. With the Englishman only feet away, Bruce turned aside, stood in his stirrups and hit the knight so hard with his axe that he split his helmet and head in two.
This mostly matches John Barbour's account of the encounter, with three significant differences. Barbour never says that the King was unarmored, and explicitly mentions that he wore a bassynet on his head. According to Barbour the king did not stand his ground, but rode straight towards his opponent. Barbour's account makes no mention of the king turning aside, but only says that de Bohun missed. Indeed, it would be difficult to turn aside by much and still reach de Bohun's head with an axe as he passed.
Other sources agree that Henry de Bohun fought Robert the Bruce, who was armed with an axe, and that Bohun died at Bannockburn. All the other details of that encounter come only from Barbour.
Here's the thing. Barbour wrote about 60 years after the battle. He probably wasn't born yet when it happened. He described his work as a romance. His patron was Robert the Bruce's grandson, and the general theme was that Robert the Bruce was Awesome, with Awesome Sauce on the side.
Here's a much more contemporary account, written within a dozen years of the battle:
On Sunday, which was the vigil of St John's day, as they passed by a certain wood and were approaching Stirling Castle, the Scots were seen straggling under the trees as if in flight, and a certain knight, Henry de Boun [Bohun] pursued them with the Welsh to the entrance of the wood. For he had in mind that if he found Robert Bruce there he would either kill him or carry him off captive. But when he had come thither, Robert himself came suddenly out of his hiding-place in the wood, and the said Henry seeing that he could not resist the multitude of Scots, turned his horse with the intention of regaining his companions; but Robert opposed him and struck him on the head with an axe that he carried in his hand. His squire, trying to protect or rescue his lord, was overwhelmed by the Scots. This was the beginning of their troubles! . .
Vita Edwardi Secundi, ed. N. Denholm Young, London, 1957.
The advanced guard, whereof the Earl of Gloucester had command, entered the road within the Park, where they were immediately received roughly by the Scots who had occupied the passage. Here Peris de Mountforth, knight, was slain with an axe by the hand of Robert de Bruce, as was reported.
Scalacronica: the reigns of Edward I, Edward II and Edward II, as recorded by Sir Thomas Gray, and now translated by Sir Herbert Maxwell, Glasgow, 1907
Gray wrote Scalacronica between 1355 and 1362, and his father fought at Bannockburn and was captured during the battle.
The Chronicle of Lanercost doesn't mention the encounter at all.
The earlier accounts are much less romantic, but far more plausible.