James K's answer, which seems cogent and self-referential enough that I'm willing to buy it without researching his points would, by the logic of his points, require it to be the center of the sun. This is mainly because the foci are mathematical points, not smeared objects in real life that have "width", so to speak, and even vary as the sun pulses.
That variability in the sun's radius itself would militate against any possibility that the surface is the intended other end of the line measured by the AU.
However, given its current definition as an exact number of meters, divorced, however slightly in practice, from being a literal "that's how far the surface, or center, of the sun is from the Earth's surface, or center" kind of definition, the easy answer is to find how far the sun's center (or surface, whichever is easier to find) is considered to be from Earth, and calculate the other distance of the two, and see which is radically closer to the currently defined distance. The logic of the folks along the way who influenced the old values no longer matters: only which is a better match to the slightly separated from any exact correlation with a "real" (whatever that could even mean given the variables vary, constantly) physical system... only which is a better match to the defined value matters to the question's answer.
Not asked, but related: it would be from the Earth's center (notional center...) as well, not from the Earth's surface.
Prallax's question in Comments for
Uwe's answer would be roughly answered by the idea that the measurement is taken from the Earth to an apparent position of the target star so that is what varies. If one decided not to have a fixed line between the sun and the star against which to relate the measured value, but rather to have a line that also moves to always be at the sun's surface (presumably the surface literally nearest to the Earth as we and the sun dance around each other during the year), then the mathematics would be hideously harder than one would desire...
So logically, the moving measurement being the Earth-to-star measurement, one would expect the sun-to-star relationship to be fixed to avoid that hideously harder math (the harder math which would benefit no one at all). Given a sensible desire to not complicate that math, one would expect it also go from the center of the sun to the center of the star so that as it moves with the Earth, it wouldn't move at all. Again, so to speak.
To reiterate, there being no value at all to complicating the math for a simple measurement, a simple system to relate the measurement to would logically include the other long leg of the triangle be fixed to as simple a set of choices as possible: center to center, sun to star.