I'll take a stab at it. First off, the concepts of open and closed circuits work well only at DC and at frequencies where the wavelength of the signal is significantly longer than the size of the circuit. At higher frequencies, the behavior of the voltages and currents becomes more complicated.
As for sending a bitstream through a pair of wires, both of your alternatives are true. The IEEE 488 bus (a.k.a. GP-IB or previously, HP-IB) uses your first method. The data transfer process uses 8 data lines and 3 handshake lines. The sender puts voltages on the 8 data lines, waits long enough for the voltages to reach the receiver, then puts a voltage on one of the handshake lines telling the receiver that the data is there. When the receiver sees the voltage on that handshake line, it measures the voltages on the data lines and determines the 8 binary values that those voltages represent. The receiver then puts a voltage on another handshake line to tell the sender that it has received the data and that the sender can send the next 8 bits. (The handshaking is a little more complicated than that, but that's close enough for this discussion.)
All that waiting takes time and limits the data rate on the bus. Also, the maximum data rate becomes lower as the distance between sender and receiver increases because it takes longer for the voltage changes to travel between sender and receiver.
Long-distance communications links and modern computer networks work more like your second method. The sender sends many bits together as a sequence of voltages on a pair of wires. The rate at which the sender can change the voltage on the wires is limited by the bandwidths of the wire pair, the sender's transmitting circuits and the receiver's receiving circuits. When the bandwidths of the components are such that the voltage can be changed rapidly, and the distance between sender and receiver is large, the sender can send many bits in sequence before the receiver has seen the first bit.
There are many ways to send binary values on a pair of wires besides just having one voltage represent a "1" and another voltage represent a "0". For example, you could use 4 different voltages, representing the binary values "00", "01", "10" and "11". On a pair of wires that allow the voltage to be changed only a certain number of times a second, using 4 voltages instead of just 2 allows you to send twice as many bit values in a given interval of time.