The word "current" contains the meaning of motion, what moves in an electric circuit is the electrons. In conductors electrons get an average drift velocity and create a current. In resistances this velocity falls very low ( that is what resistance to a current means, too) .
What causes these induced surface charges - Is it just the strength of the electric field at the terminal of the battery ? If so, do we really need the conductor to have contact with the terminal to set up this field inside the conductor?
When a battery terminal touches a wire, the field transiently propagates in the bulk of the wire, not just the surface. When a circuit has a direct field imposed on it, as battery terminals , electrons flow out of the battery and push by their drift electrons to go into the battery at the other pole.
In other words , why doesn't a bulb glow if the circuit is brought sufficiently close to the battery without actually touching it ?
If air intervenes, and the field first meets air and then the first metal of the circuit, there will be a transient current induced by the field on the conductor which will stop as the moving electrons will not be replaced, the air has large resistance.
A circuit can have an Alternating Current, AC, at the beginning, the electric field changing sign sinusoidally. Then the transient current, even through air, will be changing direction/sign and an AC current will appear, proportional to the capacitance between the source and the beginnings of the circuit. This happens because the electrons drift back and forth within the metal parts of the circuit, and do not need to cross through air. This current will light a lamp, although the capacitance will be very small between ends of wires. Capacitors have large surfaces and thus can allow larger currents . Capacitors are an integral part of AC circuits , from radios to computers , to air conditionerss, .....