I think it works like this,
Hydrostatic pressure acts in all directions, up, down, to the left, to the right, and normally on all surfaces it comes in contact with. In the teacup, the normal pressures on the wall are transmitted downwards, while in the flask shaped vessel, the hydrostatic pressures acts normally to the wall and the vertical component of that is transmitted to the floor of the vessel as a tensile load, uplifting the edge of the floor. Also notice that the weight or vertical component of hydrostatic pressure acts directly on the floor of the flask shaped vessel near the pointy part and does not act directly on the floor of the teacup, but normally along its walls.
To get the weight of water from its hydrostatic pressure you have to integrate the normal hydrostatic pressures at all points and depths over the entire surface area in which the water is in contact.
Try it by making your shapes with an infinite number of tiny cylinders stacked one on top of each other. In the case of the teacup, working from large diameter top cylinder to smaller diameter lower cylinders, at each cylindrical element's interface to the next, calculate the weight of water in the larger portion of the top cylinder's extra differential radius and add it as an axial compressive load into the wall thickness of the next lower cylinder. When when you get to the bottom cylinder, you will find that the sumation of all larger cylinder water weights causes an edge load that must be placed on the edge of lowest floor. That, in combination with the weight of the water column above the lowest floor, now the integral of which must be taken around the edge of the lowest floor equals the weight of all the water contained in the teacup.
In the case of an inverted teacup, there will be a hydrostatic "roof" load from hydrostatic pressure in the up direction, that will tend to lift the higher cylinder off the next lower cylinder. make that a tensile load on the next lower cylinder. Carry them all sequentially down to the bottom cylinder.
When you get to the bottom cylinder, both will have equal total weights of water. Pressures are hydrostatic. In the corner of the flask, the hydrostatic pressure is rho * H * density, just as it is at the center of the flask. Not the same as the teacup. In the equivalent "corner", the upper corner of the teacup, the pressure is 0.
Average hydrostatic pressure can be used, but the average of a triangle pressure is at the 2/3 point. In the case of the teacup, 1/3 from the surface and, in the case of the flask, 2/3 from the surface. Therefore there are much higher average pressures at the centroid of the inclined surfaces of the flask than there are on the centroid of the same inclined surfaces of the teacup. Over half of its surface is at less than half the flask's average pressure on the topologicly equivalent surface.
As for pressure below the vessels, underneath the steel or glass bottom, I have more pressure under my foot than an elefant, so I don't see that as too unusual that they should be different.
Now think about what happens when the vessels have no bottom, like when you take a glass of water and turn it upside down on a perfectly smooth surface. Then you have to push down hard, or the water will lift the glass off the table and run out. Another good one is a plastic sheet glued at the edges to a perfectly smooth surface, then filled with air under pressure.
"If everything seems under control, you're just not moving fast enough."
- Mario Andretti- When asked about transient hydraulics
