Traditional would mean it was folded in pleats, made of wool, in a tartan. His was definitely modern meaning sewn in pleats, made of cotton, and a solid color. Plus the design is more modern and not traditional.
What you're thinking of is "regimental" aka, "going commando". It's called regimental because the Scottish army recommended going sans underwear in order to cut supply costs, laundry costs, packing weight, dressing time, et. al. I do not know if the Scottish army still recommends this practice, but the Scottish Tartans Authority (the de facto organization about all things kilts) has actually condemned this practice, stating that "'going comando' flies in the face of decency and is childish and unhygenic."
Director Brian Wilton stated, "The idea that you are not a real Scot unless you are bare under your kilt should be thrown into the same wastepaper basket as the idea that you're not a real Scot unless you put salt on your porridge. People should not be browbeaten into believing that nonsense. Just because Highlanders wore nothing in the days before Y-fronts were invented doesn't mean that we, in the 21st Century, should wear nothing too. Sending children up chimneys is traditional too, but we don't do it now and the same should apply to wearing nothing under a kilt. Common sense and decency suggest that for the majority of occasions one should wear under the kilt what one would wear under a pair of trousers."
Also from the same article, "Kilt maker Slanj, which has provided Highland dress for Billy Connolly and Ewan McGregor, insisted last year that underwear must be worn with hire kilts. The firm, which also made a kilt for self-confessed true Scotsman Sir Sean Connery, said some kilts had been returned in such a dirty state that they were too unhygienic for staff to handle."
Source: http://www.telegraph.co.uk/news/uknews/scotland/8149592/Draught-guidance-a-kilt-need-underwear.html
I for one like the freedom, comfort, and sanitation of wearing underwear.