Where do we define the "surface" of a gas planet?

There are two common definitions in use for the surface of gas planets:

• The 1-bar surface: As pressure increases, the deeper in we go into the gas planet, we will hit a pressure of 1 bar at some altitude. Gas at this altitudes will usually sit deep enough in the gravitational well and be of a near-uniform density and temperature, as to not be influenced by exterior parameters, for example the solar wind. Therefore, the altitude of the 1-bar level will remain essentially constant, for short astronomical times.
• The $tau=2/3$-surface: This is the altitude, from which photons can escape freely into space. This happens at an average optical depth $tau$ of 2/3. It is essentially what you see in your image as the limit of the black background. For the sun one end of the photosphere is the average $tau=2/3$-surface, and for transiting exoplanets this is identical to the measured transit radius at that wavelength.

There is no hard relation between those two surfaces, but in general their altitude will not be different by more than a scale height, as at around 0.1-1 bar the gaseous atomic and molecular bands become enormously pressure broadened, which makes the atmosphere quickly opaque at most wavelengths, for the usual gas giant components.