Inside the battery

Excellent! This is correct.

Charges cannot be separated completely. Thus there are always both positive and negative charges inside the two cells. But in case of a fully charged battery there are far more negative charges.
If the cells of the battery are connected with an electric conductor, the charge distribution will be balanced out. Having said this,liquid substances do not flow from one cell to another inside a battery. Instead, only tiny charges move, namely electrons.

It is also important that the two cells are separated inside the battery, because otherwise the negative charges would already be transferred inside the battery.

In the next step, you will find out the exact movement of the charges.