What is the rower actually doing? Pushing the water or pushing the lake? A well-known example to demonstrate Newton's third law is rowing a boat on water. It can be simply explained as follows: The rower is pushing the water backwards with the oar and the water exerts a reaction force on the oar. That force is received by the entire system of boat (beacuse the rower holding the oar is inside the boat) causing the boat to move forward. But does that phrase "pushing the water backwards" mean the rower is pushing back the lake? (by 'lake' I mean the entire system including the dam as well)

Considering the confusion poped-up in the comments, I thought it will be easier to visualize the lake as a huge basin, and I think you can now understand my question. Does the oar push that basin back?
 A: Yes, the force generated by the oarsperson is applied to the Earth as a whole eventually. The force of the oars is applied initially to a local volume of water in the lake, causing it to move backwards relative to the boat. The momentum of the water propelled backwards in the immediate vicinity of the oars is dissipated into the lake as a whole and subsequently by the Earth as an isolated system.
A: During the rowing action, all that is necessary is that an increase in the boat's forward momentum is created by an increase in rearward momentum for something else.
We can easily think of this as a parcel of water.  As the oar moves, it's creating a force couple that pushes (accelerates) the rower forward and pushes the water backward.
For the purpose of moving the boat, this is sufficient.  But after this action the water interacts with its environment (the lake) and creates forces.  The result is that the parcel of water slows down and the rest of the water accelerates backward a teeny bit.
Then that water does the same to its environment.  Eventually the forces and momentum of the oar spread out to the lake and the earth.  (And then they run in reverse as viscous forces bring the boat to a halt).
So the eventual result is that the rower ends up pushing on the lake/dam/earth, but that isn't required to move the boat.  Only the acceleration of the water parcel by the oar is necessary.  The rest happens after the oar is gone and doesn't (directly) affect the boat.
A: The boatman's oars push the water backwards.
Practically, this is almost equally balanced by the boat pushing water forward (from the bows, and sideways in both directions, and pulling it forwards from the stern, creating waves) due to its form drag and the skin resistance of its wetted area.
If the boatman stops rowing, the boat stops moving forwards in a few seconds; less than a minute. So if you consider the boat and a local region of water around it as a system, very little energy escapes from that system, mostly in the form of a wake, which will eventually push equally on both sides of an idealised lake (or a 1-D lake, i.e canal); net force 0.
Assuming the lake is large compared with that local region (ignoring degenerate cases like a bathtub) there is very little interaction with the lake as a whole.
So except for the first stroke away from a dock, where he pushes aft (acceleration), and for grounding the bows on the far shore (deceleration), (and those two boundary pushes cancel out apart from the time difference between them) I suggest he does not push against the lake, only against the water.
Acceleration provides a temporary backwards transfer of momentum to the lake, while Deceleration cancels it later with forward momentum, but the vast majority of the rower's energy is dissipated maintaining a body at constant speed through the water with no net change in momentum.
