I won't argue with the wikipedia definition (although the NASA Jupiter fact sheet lists it as the radius at 1 bar), but just to point out that the scale height of the atmosphere of Jupiter is given by, $h \sim kT/m g$, where $T$ is the temperature, $m$ is the mean molecular mass and $g$ is local gravity. Putting in some numbers: $ g \simeq 24.8$ m/s$^2$, $m \simeq 2.2 m_u$ (atomic mass units), $T \simeq 165$ K, gives $h \simeq 25$ km. Thus the pressure changes extremely rapidly compared with the actual radius of a gas giant and is very small compared to the difference, for instance, between the polar and equatorial radius for a gas giant (like Jupiter) with significant rotation.
So, whether you give a planet's radius at 1 bar or 10 bar isn't going to make a lot of difference ($<100$ km).
However, your comment about whether the definition is analogous to the photosphere of a star does raise an interesting question. The effective opacity radius of a gas giant can vary with the wavelength of light. It is this property that can be used in conjunction with transiting exoplanetary systems to learn something about their atmospheres. For instance, it requires less physical depth of a gas to block wavelengths corresponding to the strong 589nm sodium doublet and thus the exoplanet is opaque at larger radii at this wavelength and thus has a (slightly) deeper transit. This provides the opportunity for transit transmission spectroscopy (e.g.
Sing et al. 2012).
Another example is that at UV and X-ray wavelengths there have been observations suggesting hot Jupiters have very extended (and opaque to high energy photons) atmospheres and may be being photoevaporated (e.g. Poppenhaeger et al. 2013).
This raises an important distinction. A definition based on the radius at a particular pressure is mostly used by theorists and solar system scientists. However, we still don't know what the atmospheric structures of exoplanets are and can't measure pressures. The radii quoted for exoplanets are more akin to how the radius of a star is measured - based on its opacity at the wavelengths being observed.