The Model 13 entered production in 1974 as an outgrowth of the 2 1/2" heavy barrel Model 10-6 in .357 magnum. The FBI issued the 10-6 but found the short ejector rod on the 2 1/2" barrel was not adequate to eject the cases from high pressure .357 Magnum loads. Thus the 3" round butt Model 13 was born as 3" was long enough to accommodate a full length ejector rod, and the 3" round butt Model 13 became known as the "FBI Model".
My Model 13-3:
It's worth noting however that the "FBI Reload" where the thumb of the weak hand is used to eject the cartridges often isn't sufficient to fully eject .357 Mag cases. The FBI reload was developed when the .38 Special was king, but doesn't work as well with the higher pressure .357 Mag. The Universal or Stress Fire reloads, where the palm of the strong hand (Universal) or weak hand (Stress Fire) are used to slap the ejector rod are more effective and more positive in ejecting .357 Mag brass.
The Model 13 was made in both 4" square butt models and 3" round butt models initially. The square butt disappeared in 1995-96, and the 3" barrel disappeared in 1997, before the Model 13 itself was discontinued in 1999. Consequently you will find 4" round butt Model 13-4 and 13-5 revolvers
The initial order for the Model 13 in 1974 was for the New York State Police, but the FBI also issued some Model 13-1s in 1974, and the Model 13 was also a popular personally owned weapon with agents.
The FBI issued the round butt, 3" Model 13-2 and 13-3 as the primary service weapon for new field agents from 1980 until 1991 when new field agents began being issued the 10mm S&W 1076. My understanding is that the last procurement from S&W for Model 13s was made in 1986-87, so I don't think the FBI issued any 13-4s, but that might not be accurate.
FBI service weapons of the time are a bit confusing as there were three FBI handguns issued simultaneously at some points, along with the other handguns still in service including previously issued Model 10-6s, and older Colt and S&W .38 Specials, as well as an approved list of personally owned handguns.
The standard issue for FBI SWAT was the S&W Model 459 from 1980 to 1988, when they switched to the Sig P-226, and then switched again in 1998 to the Springfield Armory Bureau Model in .45 ACP.
Beginning in 1983, the FBI HRT standard issue was a Novak Custom Browning High Power, and in 1995 that changed briefly to a Les Baer Custom Para-Ordnance P-13.45 SRP, before returning to the Hi Power in 1996 due to reliability issues with the P-13.45 SRP), before switching to the Springfield Armory Bureau Model in 1998, consistent with the FBI SWAT issue pistol.
When the FBI issued Model 13s were finally phased out around 2001, rather than allowing agents to buy them, the GSA determined they were "weapons of mass destruction" and required the returned FBI issue pistols to be destroyed. Consequently, the only Model 13 "FBI Model" Model 13s you'll find that were actually carried by an FBI agent will be Model 13s that were personally owned.