Oh boy... I'll try to help out on this.
The primary US Army Tank from 1942 through June 1944 were the M4 and M4A1. Both models had the Continental R975 C4 Engine; (9 cylinder, 4 cycle, radial gasoline; 400 hp). It was common that Tank Battalions had both models; parts were interchangable.
There were 3 production models (variants); all had 75mm, although the M4 was also built with 105mm in Jan 1944.
The primary difference in the variants was the M4 had a welded Hull while the M4A1 had a Cast Hull.
Early and Mid Production models had 56° Sloped Armor Front & 3-piece final drive and differential housing. Driver/Co-Driver hatches were above hoods that protruded in front of front armor.
Beginning Jan 1944, the Late production models was changed to 47° Sloped Armor Front & 1-piece final drive and differential housing. Driver/Co-Driver hatches were enlarged and inside the front armor.
These models were used by the US and UK in Africa, Sicily, Italy & France.
In July 1944, the US Army changed preference to the M4A3 with the Ford GAA Engine; (8 cylinder, 4 cycle, 60° vee gasoline; 500 hp). Early & Mid production models had 56° Sloped Armor Front & 1-piece final drive and differential housing. Late production models was changed to 47° Sloped Armor Front and Driver/Co-Driver hatches were enlarged and inside the front armor.
The M4A3 became the primary US Battle Tank, although the other variants were still in use. The M4A3 had more variants and add-ons than any other Model.
It was very uncommon to find the M4A3 in Italy before 1944, but some were used.
The M4A2 Model was only used by the US Marines in the Pacific, and Lend Lease (USSR,UK). Very few M4A2(76) were used by the Marines in the Pacific, since the 75mm was suitable against Japanese Armor. The Sherman was used in its intended role of Infantry Support rather than Tank vs Tank. There were very few instances of Tank battles in the Pacific.
Important modifications:
In Jan 1944, all M4 Turrets were modified to store Ammunition in Water or Anti-Freeze. The W was added to denote these models. Example: M4A1(75)W / M4A3(76)W.
In Dec 1944, a new track suspension system was introduced on the M4A1, and later on the other variants. These were the HVSS models designated by E8 in the description. Example: M4A1E8(76)W.
In Dec1944, the M4A1 variant was the first to receive the T23 Turret with the 76mm M1A1 Gun. By May 1945, about 50% of all Shermans in ETO were 76mm.
In Dec 1944, the M4 and M4A3 both received the 105mm M4 Howitzer in a modifed M4 Turret. It is important to note that some tank crews had already 'field upgunned' to the 105mm in 1943.
FireFly Shermans: The UK used the M4 (Sherman I-Mar 1942) and M4A4 (Sherman V-Jul 1943) variants only for the Firefly. The M4 Turret was modified by adding a radio box to the rear of the turret to allow extra room for 17pdr recoil.
According to Production Data: Some M4A3 were built into US Fireflies; none saw service.
It is important to note that some tank crews modified their M4 Turrets to install the 76mm Gun; box added to rear of turret to allow gun recoil room.
JUMBO: These were designated M4A3E2(75) & M4A3E2(76) Assault Tank; Note the E2. They arrived in France Aug 1944. Only a few were given to each Tank Battalion.
It is important to note: Patton wanted all of the M4A3E2s (Jumbos), but was denied. He had Field Engineers build "Heavies" with added armor to the front and sides. These were only the 75mm variant since there were very few 76mm M4s in the ETO. During the War, the term Jumbo was not an official name found in any document; they were simply E2s.
It is important to note that M4A3s were not issued to Tank Battalions that used the M4 or M4A1 variants. So you never saw mixed Battalions of M4A3s and M4A1s. This was due to engine & part differences.
D-DAY: You found the M4(75), M4A1(75) and M4A3(75) all in use on June 6th 1944. Firefly 17pdrs were in use by the UK on June 6 1944.
The M4A1(76) did not arrive until July, in time for the breakout.
The Jumbo arrived in late August 1944.
I hope this helps a little...