My question is, shouldn't the terminals of a battery be neutral since even though the negative end of the battery gives up an electron, it also has its positive ion leftover dissolved into the solution. Taken in isolation, yes, each electrode and its immediately surrounding solution would still be neutral.
In a battery circuit diagram, the positive and negative terminals are connected to different components. The positive terminal is typically connected to the load, which is the device or circuit that the battery powers. This allows the current to flow from the battery, through the load, and back to the negative terminal.
The extra electrons on the negative terminal can now get to the positive terminal to replace the lost electrons and will flow between the two terminals. The two terminals have different potential relative to each other. The entire battery may be any arbitrary potential relative to infinity; positive or negative.
The negative terminal is usually marked with a minus (“-“) sign or colored black to differentiate it from the positive terminal. It is designed to be the output terminal of the battery, providing the necessary electrons for the circuit to operate.
Some circuits need a negative voltage, so the positive side of a battery would be "ground". Some circuits need positive and negative voltages, in which case there could be two batteries, one with the negative side attached to ground, and the other with the positive side attached to ground. This works because voltages are relative.
These terminals play a significant role in the flow of electric current within the circuit. The positive terminal, also known as the anode, is where the electric current enters the battery, while the negative terminal, also known as the cathode, is where the electric current exits the battery.