I think the stepped ring magnum was introduced in 1899 at Rigby's behest for their 400/350 rimmed cartridge. The 350 Magnum was a later development (perhaps 1908).
The 350 magnum I owned was on a normal magnum action with the full sized receiver ring as the 416s had. The magazine dimensions were different surely though. I don't recall seeing and 350Magnums with the stepped ring.

Tthe rimmed rifles were commonly rebored and chambered to 375 H&H rather than rebarreled. I'm sure that is the case with the one pictured if we are going by the ammo in the picture. Strange though as I am not certain that the 350 Magnum can be rechambered to the H&H case without a good amount of setback and the squared breech on that barrel looks totally normal to me.