I thought that the MP38/40 was the transitional piece that took place because of the safety issues associated with the original MP38? The /40 (also I think called a II) supposedly adds the push-pull bolt and the safety notch.
I purchased one of the Inland Arms MP38's about 4 years ago. It came in with a MP38 tube, but a matching MP40 kit. It ran very poorly. Like yours, it was registered as a MP38. A friend helped me find a m/m MP38 kit which I swapped out on the tube. Surprisingly, it runs like a sewing machine on the MP38 kit. It's a sweet, very uncommon gun.