Question: Which cartridge did John Browning design that has a rim that is the same in diameter as the .50 GI?

Context paragraphs:
M2 Browning: The M2 Machine Gun or Browning .50 Caliber Machine Gun is a heavy machine gun designed toward the end of World War I by John Browning.  Its design is similar to Browning's earlier M1919 Browning machine gun, which was chambered for the .30-06 cartridge.  The M2 uses the much larger and much more powerful .50 BMG cartridge, which was developed alongside and takes its name from the gun itself (BMG standing for "Browning Machine Gun").  It has been referred to as "Ma Deuce", in reference to its M2 nomenclature.  The design has had many specific designations; the official designation for the current infantry type is Browning Machine Gun, Cal.  .50, M2, HB, Flexible.  It is effective against infantry, unarmored or lightly armored vehicles and boats, light fortifications and low-flying aircraft.  The M2 has been produced longer than any other machine gun.

M1895 Colt–Browning machine gun: The Colt–Browning M1895, nicknamed "potato digger" because of its unusual operating mechanism, is an air-cooled, belt-fed, gas-operated machine gun that fires from a closed bolt with a cyclic rate of 450 rounds per minute.  Based on a John Browning and Matthew S. Browning design dating to 1889, it was the first successful gas-operated machine gun to enter service.

Guncrafter Industries Model No. 1: The Guncrafter Industries Model No. 1 (Or M1) is a variant of the widely popular M1911 handgun, modified for the .50 GI cartridge.  John Browning's 1911 .45 ACP has been used for many cartridges over the past century.  In every instance the cartridge used a bullet with a diameter that was the same or smaller than the original .451".  Both the Desert Eagle and the LAR Grizzly have been offered in cartridges of .50 caliber.  Both are large cartridges requiring a large, heavy pistol.

.32 ACP: .32 ACP (Automatic Colt Pistol), also known as the .32 Automatic is a centerfire pistol cartridge.  It is a semi-rimmed, straight-walled cartridge developed by firearms designer John Browning, initially for use in the FN M1900 semi-automatic pistol.  It was introduced in 1899 by Fabrique Nationale, and is also known as the 7.65×17mmSR Browning or 7.65 mm Browning Short.

Remington Model 241: The Remington Arms Model 241 Speedmaster, an autoloading rimfire rifle, was manufactured from 1935 to 1951 from a John Browning design.  Remington reports that 107,345 rifles were manufactured in .22 Short or .22 Long Rifle.  A similar rifle is currently being sold by Browning Arms Company (the Browning SA-22 Browning 22 Semi-Auto rifle).

Remington Model 24: Based on a John Browning design and manufactured from 1922-1935, the Remington model 24 is a semi-automatic rifle chambered in either .22 Short or .22 long rifle.  It is very closely related to the Browning 22 Semi-Auto rifle (the Browning SA-22) which is still in production.  It is a Takedown gun, meaning that the barrel and receiver are easily separated without tools, allowing for a smaller package for transport.  A case deflector (seen in the image here) that hooked into the receiver and clipped onto the trigger guard was available.

.45 ACP: The .45 ACP (Automatic Colt Pistol), or .45 Auto (11.43×23mm) is a handgun cartridge designed by John Browning in 1905, for use in his prototype Colt semi-automatic pistol.  After successful military trials it was adopted, along with the Browning .45 Colt handgun, as the .45 M1911 cartridge.  The handgun was designated M1911 pistol.

FN Model 1903: The FN Model 1903 (M1903, FN Mle 1903), or Browning No.2 was a self-loading semi-automatic pistol engineered by John Browning and made by Belgian arms manufacturer Fabrique Nationale (FN).  It was introduced in 1903 and fired the 9×20mmSR Browning Long cartridge.  It should not be confused with the US-made Colt Model 1903 Pocket Hammerless (in .32 ACP), nor with the Colt Model 1903 Pocket Hammer (in .38 ACP).  The FN Model 1903 is based on the same mechanical design as the Colt Model 1903 Pocket Hammerless, which Browning sold to both companies (and others as well), but enlarged to handle the more powerful 9mm Browning Long cartridge.  Due to its reliability, accuracy, light weight, and quick reloading, the M1903 was an issued sidearm for many police forces and militaries.  The pistol was initially introduced by FN as the Browning Modèle de Guerre (Browning War Model) or Browning Grand Modèle (Browning Large Model).

Caliber conversion sleeve: A caliber conversion sleeve or adapter sleeve is a device which can be used to non-permanently alter a firearm to allow it to fire a different cartridge than the one it was originally designed to fire.  The different cartridge must be smaller in some dimensions than the original design cartridge.  Alternative names sometimes imply the type of dimensional difference.  A chamber insert may be used for a shorter cartridge of similar base diameter.  A supplemental chamber or cartridge adapter is typically used for a shorter cartridge of reduced diameter.  A cartridge conversion sleeve may include a short barrel of reduced bore diameter.  Shotgun conversion sleeves may be called subgauge inserts, subgauge tubes, or gauge reducers.  Sleeves intended for rifle or handgun cartridges may have rifled barrels.  Additional variations may allow centerfire weapons to fire rimfire ammunition and/or retain autoloading function with the smaller cartridge.

.50 GI: The .50 GI (12.7×23mmRB) pistol cartridge was developed by Vic Tibbets and Alex Zimmermann of Guncrafter Industries.  The .50 GI was introduced at the 2004 SHOT Show alongside the Guncrafter Industries Model No. 1, a variation of the M1911.  The round has a rebated rim that is the same diameter as that of the .45 ACP.

Supporting paragraphs: .50 GI .45 ACP
Answer: .45 ACP