1-out-of-2 OT, chosen 1-out-of-2 OT and Random 1-out-of-2 OT

The three are closely related. At a high-level, the difference is with respect to the input/output behavior of the three, but in fact they are ultimately all equivalent functionalities (a construction for each can be used to to create any of the others [1]).

• 1-out-of-2 OT: In this case, Alice inputs two bits, $b_0$ and $b_1$, and Bob inputs nothing. Bob receives two bits, $c$, chosen with uniform probability, and $b_c$. Crucially, he doesn't choose the value $c$, it's an output of the functionality [1].
• Chosen 1-out-of-2 OT: Alice again has inputs $b_0$ and $b_1$, but now Bob can select $c$ himself, and receive $b_c$ [1].
• Randomized 1-out-of-2: In this case, neither Alice nor Bob have inputs to the protocol. Instead, they interact as prescribed by the protocol, and as a result, Alice receives bits $b_0, b_1$ both chosen uniformly at random, and Bob learns $c$ and $b_c$ [2].

[1] Precomputing Oblivious Transfer by Donald Beaver, 1995

[2] Randomized Oblivious Transfer for Secure Multiparty Computation in the Quantum Setting by Bruno Costa, Pedro Branco, Manuel Goulão, Mariano Lemus, Paulo Mateus