The H1 reference is a typo commonly repeated across the Internet. The round should actually be referred to as Hl/A, B, C etc the Hl which is short for Hohlladung, the German for Hollow Charge or HEAT as it is now more commonly referred to.
This question has been debated on several other forums but unfortunately no definitive date for the introduction of the different variants has been identified, at least not in western English language sources as far as I'm aware. The information I have suggests;
The 7,5 cm Panzergranate 38 (HL) was made available in June 1940. The improved versions A, B and C followed, beginning at the end of 1941 and onwards into 1943.
Source: Die deutschen Sturmgeschütze 1935-1945 by Wolfgang Fleischer. Dates for the production of the subsequent versions is given by the same source as late 41 for the "A".
In Steven Zaloga's book Armored Champions he has stated that the 'A' variant was first delivered to Eastern Front units in January 1942 with later versions introduced throughout the remainder of 1942. Best guess for the 'B' would be mid to late 1942.
Hope this helps
"Whoever wishes to master the art of war must study it continuously. I....am of the opinion that one lifetime is not enough to attain this goal." - Frederick II