How close does the wire have to be before it suddenly becomes a capacitor? If it's moved just a little bit further away, why does it suddenly stop being a capacitor?I don't believe that electrons will leave the negative terminal of a battery until new electrons enter the positive terminal of the battery. Positive charges within the battery hold the negative charges inside the battery.
Connecting a long wire to one end of the battery does not, in any real sense, make a capacitor. No current will flow into the wire. Positive charges within the battery would prevent that from happening. The wire becomes a capacitor only when it is held close to the positive terminal. With this additional provision, there would be a small amount of current flowing for a short period of time.
I suggest that the original poster see A Kitchen Course in Electricity and Magnetism by David Nightingale and Christopher Spencer.
If our battery is 1.5V (I don't recall if the OP ever put forth a voltage), then even 1 aF (one attofarad, which is one one-millionth of a picofarad) capacitance would be enough to require ten electrons to charge it to that voltage. Just 0.1 pF would be enough to require a million electrons.
Now consider that two one-inch long pieces of 24 AWG wire separated by 6 cm (the height of a D cell battery) have more than 0.1 pF of capacitance between them.
Also, as previously described, if no charges are transferred to the wire, then you have a conducting wire that is in the presence of an electric field having a tangential component but without any resulting current flow.