How did the US Transit Without a Visa program work before 9/11?

According to a DHS press release from 2 August 2003 announcing the suspension of the TWOV and ITI programs effective on that date:

The Transit Without Visa program has been in use in the United States since 1952. It applies to passengers who normally would be required to obtain a visa to travel to the United States. Under the TWOV program, passengers arriving in the United States from a foreign country are permitted to travel through the United States to another foreign destination without first obtaining a visa to stop and change planes in the United States. Passengers under the TWOV program go through the full border inspection process upon arrival in the U.S. Under the TWOV program, a passenger may stop at one or two U.S. airports en route to another foreign destination. If on a domestic flight to a second U.S. airport, the airline is responsible for ensuring that the passenger does not illegally enter the United States. Airlines provide contract security escorts and are required to maintain control of the passenger’s passport and other travel documents.

The International-to-International transit program also allows passengers arriving from foreign countries to transit through the United States to another foreign destination without first obtaining a visa. Unlike the TWOV program, however, ITI passengers may only transit through one airport, and they may not leave the international transit lounge while connecting planes at that airport.

In 2003 when the program was suspended, the relevant regulations (8 CFR 212.1(f)) said:

(1) Transit without visa. A passport and visa are not required of an alien who is being transported in immediate and continuous transit through the United States in accordance with the terms of an agreement entered into between the transportation line and the Service under the provisions of section 238(d) of the Act on Form I–426 to insure such immediate and continuous transit through, and departure from, the United States en route to a specifically designated foreign country: Provided, That such alien is in possession of a travel document or documents establishing his/her identity and nationality and ability to enter some country other than the United States.

(2) Unavailability to transit. This waiver of passport and visa requirement is not available to an alien who is a citizen of Afghanistan, Angola, Bangladesh, Belarus, Bosnia-aherzegovina [sic], Burma, Burundi, Central African Republic, People’s Republic of China, Colombia, Congo (Brazzaville), Cuba, India, Iran, Iraq, Libya, Nigeria, North Korea, Pakistan, Serbia, Sierra Leone, Somalia, Sri Lanka, and Sudan.