I do not believe in the rhetoric from the video that “hydrogen bombs” are a “thousand” times more powerful than mere “atomic” bombs. The yields pure fission devices overlap with the yields of some thermonuclear warheads.
A thermonuclear weapon is invaluable because most of its yield is derived from the secondary, as it does not require fissile isotopes (which would be difficult to procure), but from common deuterium and lithium and the fast fissioning of the uranium tamper from the fast neutrons produced by the fusion of the lithium deuteride.
I do not know precisely the technical difficulties of a thermonuclear weapon, but one would involve making sure that the secondary does not prematurely explode. From a technical perspective, one should appreciate how difficult it is for the bomb designers to model the neutron flux from the exploded primary to the secondary (as those neutrons could prematurely trigger fusion of the lithium-deuteride) before the tamper is compressed ( as the tamper ablated away from the intense electromagnetic radiation from the primary).
Does North Korea want to be credible? They could give a rough estimate of the yield from the primary device if it successful and the theoretical yield from whatever secondary their bomb had. The sum of these two estimates could be compared with the yield from the nuclear test. If the test has a yield higher than what was expected from the primary alone and is congruent with the theoretical yield of the thermonuclear device, one can make an inference that they have some mastery of thermonuclear weapons.
I do not see a compelling reason for North Korea to lie about the test. A nuclear deterrent should be as credible as possible in order to deter aggression away from them. If North Korea is perceived to exaggerate its nuclear arsenal, its defensive capabilities would be dismissed.
What is a “hydrogen bomb”? It has to be something more than using some deuterium-tritium mixture to boost the primary. Maybe it did not use separate stages, but had a layercake design, like RDS-6, with a layer of lithium-deuteride outside the primary.