A plumb line defines local vertical
The gravity direction at a location defines its vertical. A tangent plane perpendicular to it is local horizontal. On a spherical approximation, verticals converge toward the centre and horizontal planes turn with position. Calling each plane level does not make them one infinite plane.
Why density does not choose a direction
For an immersed volume V, buoyancy has magnitude ρfluid gV, opposing gravity. Weight has magnitude ρobject gV. Their difference depends on densities, but both terms contain the acceleration and its direction. With no external acceleration, a density comparison alone does not produce a downward force.
A surface can be smooth and curved
Water in static equilibrium follows an equipotential. Tides, currents, wind and pressure add departures from it. A short water level is an excellent local reference; extending it over kilometres requires attention to curvature and refraction.