The Danish axe (long hafted) is for use on foot and required skill to wield, hence it was used by experienced Huscarls and household infantry. Cavalry, particularly lighter cavalry in the Middle East and Asia evolved 'saddle axes', which are short hafted and designed to be swung underarm at infantry (kinda like a polo swing with attitude) which introduced the blade usually just under the chin in an uppercut with gruesomely effective results....
The illustration is of a commander who was more likely to dismount to fight. The illustrator has obviously seen the image of William and Odo on horseback with their Mace and Club resting on their shoulders depicted in the Bayeaux tapestry...
I agree the sword was a higher status weapon and would have been favoured more by higher ranking knights...