If you must think in terms of electrons, then you are sufficiently correct for your purposes. Just hook up the wires from the tissue to the op-amp inputs, and all will be well.
The electrons are confined to the wires laterally, they don't move sideways across the insulation, they are like pipes in that respect. At the ends, in contact with the op-amp or the biological material, they do move in and out of the ends of the wires, again like pipes fill and drain from their ends.
What goes on at the tips is rather complicated. If a current flows at the interface, then oxidation and reduction reactions occur changing the Na between +ve ion and neutral metal, while electrons are exchanged with the respective wire tips in the process (see electrolysis). Typically though, no significant current will flow into the high impedance op-amp as the wires will charge up very quickly. Charging stops when the tips reach an equilibrium voltage depending on the concentration of the various ionic species in their vicinity.
With only two electrodes, we only see the difference in ionic concentration and so voltage between the two tips. If we had a third remote electrode on the body, a leg-drive for instance, then we could take this as a 'ground' for the receiving equipment, such a connection is often used to control the common-mode voltage at the op-amp inputs. Without this third connection, the two wires you show will still faithfully transmit the differential voltage at the wire tips to the op-amp.
This is exactly the same principle as batteries driving a signal down the wires, the only difference being the magnitude of the signal. A chemical battery (lead acid, alkaline, lithium, whatever), they all work with oxidation and reduction reactions happening at the electrode electrolyte interface, shifting electrons into and out of the ends of their connected wires.
Your graph of the electron distribution along the wires is wrong, but it's not intuitively obvious given your apparent level of understanding why it should be wrong. Fortunately, that graph doesn't matter, so you can safely ignore it. As engineers, we just tend to think in terms of voltage and current, and leave electrons to the solid state and plasma physicists.
The differing ionic concentrations in the tissue do give rise to a potential difference between the tips.
There are two better ways to describe what happens after that.
One is to regard the pair of wires as transmitting the voltage between the tips to the op-amp end. If you can ignore how they do this, all well and good, they just do.
The other is the hydraulic analogy, water in pipes. A lot of people don't like this model for electricity, as it's so obviously wrong. Wrong as it may be, it is very useful. What it does do is give you a model for voltage, as pressure, which messing with electrons doesn't do until you are a lot more advanced. Consider each wire as a pipe for electrons, just like a water pipe is a pipe for water. With no or low flow, the pressure or voltage is constant along the pipe or wire. Electrons or water moving out at one end are automatically replenished at the other. This doesn't happen by law, or by magic. If they are not replaced, then the pressure or voltage of the wire or pipe becomes so overwhelmingly large with respect to its connections that the replenishment is forced to happen. Actually, it happens so reliably that it has been turned into a law, Kirchoff's Law
