What is exposed in the surface tension of water?
Surface tension may be described regardless if all / some / none of the water molecules expose one or both hydrogen atoms or the oxygen atom toward air, even if there are studies in this field (e.g., example, example).
Key points however are

the movement (translation, vibration) of the water molecules in the bulk of water droplet in restained, because of intermolecular attractive forces (e.g., hydrogen bonds), as well as intermolecular repulsion (molecules normally do not come closer to each other than the sum of the corresponding van der Waals radii).  Like with humans, there is an optimal intermolecular distance where overall the energy passes a minimum.
In extension of your drawing of the situation in the water droplet, I add the following figure where molecules are reduced to a black sphere; these are either in the inner of the droplet (light blue background), or at the interface (dark blue background) exposed to air.  The black arrows indicate intermolecular attractive forces (responsible for cohesion).

(derived from wikipedia)

For a molecule in the inner of the droplet, surrounded completely by molecules of the same type, the net force on this molecule is zero.

For a molecule at the interface water / air, the attractive forces to molecules of water in the inner of the water droplet exceeds the attractive forces between this water molecule and air molecules.  Thus, there is a net force pointing to the inner of the water droplet; this is symbolized by the red arrows.
Because of this inward directed force, the droplet thus tends to contract, building an inner pressure until compensated by the earlier mentioned intermolecular repulsion.


Overall, the lowest energy of the water droplet would be achieved as a sphere.  Sometimes, this is shown in microgravity (Scott Kelly at 8:28min).  On earth, however, things like gravity, contact with a surface (adhesion), relative air speed (e.g., if a drop falls down the rain gutter) etc. lead to deviations from this ideal.  However no, because the molecules in the inner of the water droplet are the same as the ones at the interface exposed to the air, there is no difference in the intermolecular, pairwise attractive forces between molecules of same type.