How does the current know how much to flow, before having seen the resistor?
Not sure if this is what you're asking, but yes, when the battery is connected, an electric field wave travels from the battery down the wires to the load.  Part of the electrical energy is absorbed by the load (depending on Ohm's law), and the rest is reflected off the load and travels back to the battery, some is absorbed by the battery (Ohm's law again) and some reflects off the battery, etc.  Eventually the combination of all the bounces reaches the stable steady-state value that you would expect.
We usually don't think of it this way, because in most circuits it happens too quickly to measure.  For long transmission lines it is measurable and important, however.  No, the current does not &quot;know&quot; what the load is until the wave reaches it.  Until that time, it only knows the characteristic impedance or &quot;surge impedance&quot; of the wires themselves.  It doesn't yet know if the other end is a short circuit or an open circuit or some impedance in between.  Only when the reflected wave returns can it &quot;know&quot; what's at the other end.
See Circuit Reflection Example and Transmission line effects in high-speed logic systems for examples of lattice diagrams and a graph of how the voltage changes in steps over time.  See Termination of a Transmission Line for an animated simulation of different terminations that you can modify, and this for a light switch example.
And in case you don't understand it, in your first circuit, the current is equal at every point in the circuit.  A circuit is like a loop of pipework, all filled with water.  If you cause the water to flow with a pump at one point, the water at every other point in the loop has to flow at the same rate.
The electric field waves I'm talking about are analogous to pressure/sound waves traveling through the water in the pipe.  When you move water at one point in the pipe, the water on the other end of the pipes doesn't change instantly; the disturbance has to propagate through the water at the speed of sound until it reaches the other end.