I just talked to Hank Rotoff about this, the H was the predecessor to the AH series. The AH series have an improved cylinder, the port belt area below the exhaust port and the transfer port area was increased in strength, also the crankcases were improved. The H82 verses the AH82 have a larger intake transfer port hole which was a weak area, the AH82 had that transfer port hole a little smaller with increased aluminum in the casting around the transfer port area to increase the strength of the barrel. Hank said to look at the outside of the cylinder below the exhaust port, the H82 is cast straight where as the AH82 has more material and tapers out to the cylinder mounting boss. I know the later AH58's have a heaver cast cylinder than the earlier AH58's, that took place around mid 1961. Power Products are notorious for spliting the cylinders in the port area, that is why I run girdles on my engines. All parts are interchangable between the A and AH series. If I knew how to post pictures here I would do the article on the improved AH58 from KART magazine.
As to why they went from H to AH, it may be they needed a way to identify the single from the twin. By the service manual I have they also built an AH and AV80, 8.0 cubic inch. The twins were BH and BV60, 6.0 cubic inch and the BH and BV69, 6.9 cubic inch.