Irrespective of what Modern Railways suggest, just a cursory glance at the two vehicles will tell you that they are not the same. For starters, the Mk4 has a tilt profile that the Cl156 lacks. They may have shared a number of construction techniques, but that does not make them the same. Likewise the Mk4 isn't derived in any particular way from the APT except in the sense that the IC225 concept was very similar to the APT-S.
The Mk3 based MU's really were based on the Mk3 coach, and you can see the similarities much more clearly. In fact, they just modified the Mk3 coach construction jigs in order to start turning out MU's. However, I'm not sure that it was a good idea, as I'm fairly certain that it introduced a number of weak spots into the structure that would be tested in the event of a heavy collision.
I have definately seen pics of a Cl317 driving trailer that had been in a heavy collision with a Peak during a shunting accident at Cricklewood in the 1980's and the coach had been bent forward of the leading set of passenger doors. This obviously will have compromised it's structural integrity so it was scrapped and another built with the same number. Now I believe the two Cl150 vehicles that were written off had been in collision with a bridge (St Helens) and a digger or some other heavy road vehicle (Seamer), so it's not beyond the realms of possibility that they suffered similarly heavy damage. I do recall seeing a pic in a magazine of one of these vehicles being scrapped, but I can't bring to mind enough detail to confirm the extent of damage.
O L Leigh