Depending what kind of reading material you're looking for (e.g. high level details or mathematically/jargon dense papers) look for things discussing "Rosseland optical depth" and "grey atmosphere approximation".
At a high level, the common convention is to define the radius by finding where the optical depth is 2/3 when using Rosselands clever way of calculating a weighted mean of the opacity from that layer in the star to space across different wavelengths of light. 2/3 being a clever derivation from Eddington where, in an idealized model of a star, that's when the actual temperature of the star should equal its blackbody equivalent temperature.
Pedantically, this distance to the point of equality is an ever so slightly different value than the "distance from the center where there is a 50% chance a photon traveling directly outward will escape the star without another interaction" rule of thumb (in the same idealized grey model). Practically, that the difference is so small is why it's a fantastic rule of thumb explanation.
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Amateur stargazer's understanding:
Depending what kind of reading material you're looking for (e.g. high level details or mathematically/jargon dense papers) look for things discussing "Rosseland optical depth" and "grey atmosphere approximation".
At a high level, the common convention is to define the radius by finding where the optical depth is 2/3 when using Rosselands clever way of calculating a weighted mean of the opacity from that layer in the star to space across different wavelengths of light. 2/3 being a clever derivation from Eddington where, in an idealized model of a star, that's when the actual temperature of the star should equal its blackbody equivalent temperature.
Pedantically, this distance to the point of equality is an ever so slightly different value than the "distance from the center where there is a 50% chance a photon traveling directly outward will escape the star without another interaction" rule of thumb (in the same idealized grey model). Practically, that the difference is so small is why it's a fantastic rule of thumb explanation.