The idealized answer is: for a well-corrected lens, the plane of best focus is intended to be flat and parallel to the sensor. That’s why standard depth-of-field diagrams usually show flat planes.

In real lenses, though, the focus field can be curved. This is called field curvature, a lens aberration where points near the edges reach best focus at a different distance than points in the center. A simple single lens tends to produce a curved field, while more complex lens designs use corrective elements to make it flatter.

So your first diagram is the practical approximation for most modern lenses; the second reflects what can happen with field curvature, though the exact shape depends on the lens design rather than being a perfect sphere in a simple diagrammatic sense.

Yes, the shape can be altered optically: lens designers can reduce or leave some field curvature depending on the intended rendering. Historically, film could sometimes flex slightly to better match a curved image field, but digital sensors are rigid, so flatter correction is more important today.