The pressure is defined as the flow rate of x-momentum in the x-direction plus the flow-rate of the y-momentum in the y-direction plus the flow rate of the z-momentum in the z-direction divided by three. Each component of momentum is conserved, and flows locally from point to point in a fluid. Each component is like a charge, and has it's own current, which makes a tensor of stresses.
A fluid does not sustain shear, and this is true whether it is still or moving, by the principle of relativity. This means that if you put fluid between two plates, and squeeze, the force per-unit-area with which you squeeze (the local flow of momentum in the direction perpendicular to the plates) is equal to the force per unit area pushing outward at the edge of the plates. The flow of momentum is the same in all directions.
You can measure the pressure by putting a homogenous solid at the point in the liquid, and noting how much it compresses. You can also do it by noting the very slight change in density of the fluid with pressure.
The pressure of fluids is not an exact description of the fluid when there is viscosity, but it is close to perfect, and the viscosity and the pressure are independent ideas which can be treated separately.
For the purpose of your class, you can think of the pressure as the amount a tiny box-spring will compress if you place a scale model in the fluid, moving along with the water. This pressure is the same in all directions in the fluid, despite the answer you got earlier.