Current flowing inside battery/voltage source Current is the flow of charges (many say electrons).
Do these charges flow just between the terminals of the source/battery (terminal to terminal) ?
Or
Do these charges flow through/inside the source/battery as well?
What I know is that the positive terminal is the lack of electrons as compared to the negative terminal which is the excess of electron.
Current flows when electrons flow from the negative terminal of the battery to the positive terminal of the battery/source?
My question is what happens when the electrons reach from the negative terminal to the positive terminal? Are electrons then "absorbed" by the positive charges or they continue to flow through the source/battery to move in cycles?

 A: In a generator the electromotive force, produced by wires moving in a magnetic field, pushes the arriving electrons down the wire to the negative terminal.  In a battery, positive ions pick up the arriving electrons which had been previous given up to the negative terminal.  (Ask a chemist about that.)
A: What happens to the electrons in the source depends on the type of source. 
Generator:
In the generator, the magnetic field exerts a force on the electrons which pushes them from the positive terminal to the negative terminal. Hence, in the generator source, the electromotive force pushes the electrons in the wire from the positive towards the negative terminal. 
Battery:
The battery has two different electrodes: the anode and the cathode. If the battery works as a source, the positive terminal is the cathode and the negative terminal is the anode. The electrons are more stable in the cathode than in the anode. Therefore, electrons will travel from the anode through the electrical circuit towards the cathode. In the anode, the electron leaves behind an ion and in the cathode, the electron combines with an ion. Hence, in the battery source, electrons leave the negative terminal and enter the positive terminal but they do not travel between the terminals. In the battery source, the ions will travel between the two terminals but the electrons not.
