Why liquids in hydrostatic equilibrium stay at the same height? (open tubes) 
This part of the book says that, because the two liquids have the same pressure at the same height (that's true), then if a liquid has a higher height, the liquid would have more pressure at a horizontal line, and then the liquid would tend to flow to the weaker side. However, it isn't really pressure that drives the liquid, it is force (or acceleration). So the flow of a liquid shouldn't depend of the pressure, but the force at that line. And the force, depends of the diameter of the tube, because force is dependente of pressure. So in this case, the two tubes have diferente diameters, therefore the acceleration that drives the liquid from one place to another would depend of the diameter. Then we couldn't just say that the liquids tend to equate.
 A: Try imagine a thin wall placed in the center of horizontal tube to prevent the flow between two sides. You agreed that $P_{right}>P_{left}$, then try to calculate the total force acting on this imaginary wall. The surface of this wall on two side is the same but the pressure on the right is larger than the left. Total force will point toward left side, this make water tend to flow from the right side to the left side.
A: According to the given fig.13.6, the liquid is at height $d_1$ in the narrow tube and at height $d_2$ in the broad tube. So, naturally(as you say) there is a pressure difference at the lowest point of the tubes. Let $P_1$ and $P_2$ be the two pressure at the lowest point of the narrow tube and broad tube respectively. And we also know that $P=\frac{F}{A}$, where A is the cross-section of the tube.
We also know that here the diameter of the horizontal tube is constant i.e. the cross-section of that tube,A is const.
So force acting on the liquid in the direction of the broad tube is $F_1=P_1.A$ and in the direction of the narrow tube is $ F_2=P_2.A$, where $A$ is the cross-section of the horizontal tube. Now since, $P_1>P_2$; $F_1>F_2$ , i.e. there is a net force on the liquid in the direction of broad tube. That means there is a acceleration of the liquid along the horizontal tube in the direction of broad tube which makes the liquid to flow from narrow to broad tube. So as you said, "....the flow of a liquid shouldn't depend of the pressure,..........." is not right.
