What are we seeing when we see the curvature of Earth?
the horizon is a circle centered at the viewer's position


There are two things wrong with this. The first is the assumption that the Earth is entirely spherical - there are no hills, mountains or other bulges to block the view. But let's make that assumption, because there's a more fundemental misunderstanding in the geometry.

The horizon is a circle that is centered horizontally at your position, but not vertically, because by being above the surface you can see a little way around the curve of the planet. 

If you're standing on the ground you are only looking from a couple of metres altitude, so you can't see very far around - this means that the circle of the horizon is centered only a small distance below your feet. Note that from this height you can't make out any curvature in the horizon.

If you're standing at the altitude from which your picture was taken, the horizon circle is centered some depth into the planet. So you are not looking out across a circle, as you imagine - you are looking down, onto a circle. Or if you like, you're looking down onto a disc that happens to have some portion of the planet piled on top of it. And when you look at a circle (or a disc) from above, it appears curved.