...meaning there are ions in both plates which altogether have a
Ions are defined as "atoms or molecules with a net electric charge due to the loss or gain of one or more electrons". There are no ions on the plates of the capacitor. There are fixed protons and free electrons constantly moving (thermally) about the positive charge in a kind of "electron cloud}".
Diagram of "uncharged capacitor"
Your diagram of the "uncharged" capacitor is misleading. You show the positive and negative charge lined up. In reality they are randomly oriented. See Fig 1 below.
When you connect an uncharged capacitor to a battery, the electrons
from the battery’s anode move to the capacitor's plate connected to
the anode (plate A on the illustration below), whereas electrons from
the other plate of the capacitor (B) move to the battery’s cathode.
Better to say that the cathode pulls free electrons off of one plate, making it positively charged, and the anode deposits an equal amount of electrons on the other plate, making it negatively charged. The battery does not supply electrons and protons. It supplies the energy to move electrons from one plate to the other. See FIG 2 below.
Your diagram of the charged capacitor is incorrect. It shows a total of 4 more electrons and 4 more protons than were on the uncharged capacitor.
It appears you are under the incorrect impression that the battery supplies electrons and protons to the capacitor. It does not. It only performs the necessary work to move electrons already existing in the capacitor from one plate to the other.
Does this mean that the battery is not moving electrons from one plate
to the other, rather it’s only moving charges?
The only charges that can be taken off one plate and deposited on the other are electrons. The protons are fixed in place and cannot move.
Does this mean that all electrons are removed from the ions on the
plate (B), and all that’s left on the plate (B) are cations, whereas
the plate (A) now has free-floating electrons from the battery, as
well as the previously present ions?
I'm not sure I understand what you are asking. The amount of net charge $Q$ on the capacitor will depend on the capacitance and battery voltage according to
Hope this helps.