Fundamentals: The role of Battery in a circuit As I understood, the analogy of a charge moving through a circuit is similar to water flowing from a high altitude to a low altitude (like waterfalls).
But in waterfalls there are two requirements if we wanted to keep that same water repeating the cycle: we need a gravitational field (which is caused by the earth),and an energy supplier that will do work on water to move it from the low altitude (low gravitational potential) to high altitude (high gravitational potential).
In circuits, I do understand that the battery does the part of moving a positive charge from a low potential energy point (the negative terminal) to the high one (the positive terminal) so it can -again- move to the negative terminal naturally.
But for that movement to occur (moving in the external circuit) we need an electric field (in analogy of the waterfall we need earth to establish its gravitational field), so the battery should establish an electric field, and give energy to charges in the wires to "re-climb" to higher potential energy points relative to that field?
 A: The hydraulic analogy is, in my opinion, not a good analogy. The main problem with the analogy is that hydraulic flow is substantially more complicated than a circuit. I want analogies to be simpler than the thing they are representing, not more complicated.
You have identified one such complication. Indeed, this is a difference between the hydraulic analogy and a circuit. In a circuit the “height of the waterway” depends on whether the battery is connected or not.
My recommendation is, since the analogy is confusing you, abandon it. Learn circuits on their own terms using their own equations, which are far simpler that the equations of fluid flow. You do not need the hydraulic analogy to understand circuits. They can be understood directly. There is no benefit in patching up an analogy when the time spent fixing the analogy and then using it to understand circuits could instead be used to directly understand circuits.
A: When battery moves a positive charge, battery also moves the electric field that exists around the positive charge. All the electric fields around all the positive charges that the battery moved to the positive terminal form an electric field around the positive terminal.
