Inside the battery

No, unfortunately your assumption is not correct.

There are two facts which are arguments against this version.
On the one hand, the negative charges always move (electrons). The positive charges don't move into the other cell of the battery. Charges don't mix together like liqiuid dyes.
On the other hand, 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.

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.