Inside the battery

No, unfortunately your assumption is not correct.

Indeed, most battery symbols are represented that way, but there are two facts which are arguments against this version.
On the one hand, the negative charges always move (electrons). The positive charges never move into the other cell. On the other hand negative charges neither can be substituted nor be replaced by positive charges nor negative charges turne into positive charges.

Please note too: 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.