Does the weight of fluid in a conical container act entirely on the base?

No, the entire weight will not directly rest on the base of the slanted container (although it does indirectly). There are a number of ways to approach this, but the easiest way is to observe that the total force acting on the bottom of the container is equal to the sum of the hydrostatic pressure force (the pressure at the bottom of the container multiplied by its area) and the shear force around the edge of the base (draw a free-body diagram to convince yourself of this). The hydrostatic pressure depends only on the height of the column of fluid above the given location ($P=\rho gh$), and the shear force is equal to the weight of the fluid outside of the base (supported by the walls). Since both vessels contain the same volume, but the slanted container has a wider cross section above the base, the total height of the body of fluid will be less for the container with slanted walls than the other container. Thus, the hydrostatic pressure at the base will be less. But remember, any reduction in the pressure force will be compensated by an increase in the shear force around the edge of the base. The vertical walls support no vertical force, and so they exert no shear force on the base. The slanted walls do bear some of the weight, and so they transfer this force to the base via shear. In both cases, the sum of the hydrostatic pressure force plus the shear force (if there be any) is equal to the weight of the fluid, and this is what the scale measures.

As Rahulgarg mentioned, the pressure does not depend on the shape but on the depth. However, the direction of the force caused by pressure can be approximated as being normal to the surface, hence the total force on the sides will depend on the shape. For a fluid at rest like the one I think you are assuming, the pressure at any depth will be $p=p_0+\rho g h$, where $p_0$ is the atmospheric pressure at the surface, $\rho$ is the fluid density, $g$ the acceleration of gravity and $h$ the distance from the liquid's surface to the depth you want to know $p$. In your specific example the pressure at the bottom of case 2 will be smaller than in case one because of the different heights of the vessels.