The Astronomical Unit (AU) is more than just an answer to the question "How far away is the Earth from the Sun" - it also provides a vital tool for surveying the universe, because it offers a fixed unit with which to compare vast distances across space. And because the actual distance between the Earth and the Sun varies, astronomers have had to come up with a way of settling on a fixed figure for the AU.
You might wonder why we need an AU at all - couldn't we just measure the distance in metres? The need arose from the poor knowledge of the gravitational constant, G. Even today it is only known to about a part in 100,000 (with a corresponding uncertainty in the mass of the Sun) whereas the positions of the planets are known, thanks partly to the space programme, a million times better. Calculations in celestial mechanics are therefore performed in solar masses and astronomical units, rather than in SI units (kg and m) which would carry along the 1 in 100,000 uncertainty.
As far as the transit is concerned we have used the original geometrical definition of the radius of the Earth divided by the angle (in radians) that the radius would make as seen from the Sun. Given the difficulties of measuring transits that is quite adequate. For other purposes a much tighter definition is required.
Perhaps more importantly, the actual distance of the Sun is changing continuously. The Earth's orbit is not quite circular (the eccentricity= 0.017) and the centre of the Sun is not quite the centre of gravity of the Solar system, so what does "the Earth-Sun distance" mean? The elliptical orbit is important because only an inverse square decrease of gravitational force with distance will give a closed elliptical orbit, and the recognition of this fact allowed Newton to understand the Solar System dynamics for the first time, rather than simply describing them. The centre of gravity of the Solar system as a whole depends on the position of all the planets, which is why it is not quite at the centre of the Sun, so the Sun wobbles slightly to the sides of the orbit of the centre of gravity round the Galaxy. This is the wobbling which has recently been turned to very good use in discovering other planetary systems. (There is a list of other details, for instance the Sun loses 4 million tonnes of mass converted into radiation each second, and more in the mass of the solar wind, so the Solar system is inevitably slowly expanding. And then, there is general relativity, but perhaps that should be left for another time.)
The AU has two roles to play: to give the actual distance to the Sun, and to serve as a fixed, unvarying unit for surveying the Universe. The two roles are incompatible since the actual distance is not constant. The ellipticity problem can be tackled by defining a mean over the year, but the mean is then neither the major axis nor the minor axis, since the Earth spends less time in the half of the orbit closer to the Sun (where it its speed is greater) than in the other half.
The modern definition of the astronomical unit is not, in fact, the radius of the Earth divided by the angle it subtends at the Sun, but:-
"the radius of a circular orbit, in which a body of negligible mass, and free of perturbations, would revolve around a body whose mass is one solar mass unit in [2 x pi/Gaussian constant] days." (The days are in "barycentric dynamical time" but that need not concern us.)
The mean year is then 365.2568983 days which yields 149,597,870,691 metres for the AU.
( Roughly, that translates as the distance of a planet taking exactly a year to complete a perfectly circular orbit round a star of the same mass as the Sun)
Ignoring the details and big numbers, we have reached the crucial point. The Gaussian constant, and therefore the AU, is a number decreed by the International Astronomical Unit, it is not a measurement. (It is not the only constant-by-decree; the same is true of the speed of light in a vacuum, it is now a defined number. It will not change even if someone finds a way of measuring it to 20 places of decimals.)
It is worth going one detail further. The defined value of the Gaussian constant is 0.01720209895 (AU)1.5. Ignore the number and look at the power of 1.5. This comes straight from Kepler's third Law T2 is proportional to R3 i.e. T is proportional to R1.5, so the definition is based on Newtonian dynamics, which are well known to differ from the more accurate General Relativistic treatment. The definition still needs further refinement!
This may sound like angels-on-pinheads but it is not "just theoretical" if you are trying to land a spacecraft on a comet after a 10 year chase or, I suppose, if you are an astronaut hoping to find the only landing strip on Mars - then, calculations like these would be vital.