planet - How precisely is planetary tilt defined (the tilt direction, not just the angle)?

A bit too simply (but not too much), it's the angle between the planet's rotational angular velocity vector and the planet's orbital angular momentum vector.

Where it gets tricky is dealing with little wobbles and such. Planets don't quite orbit in a plane because of gravitational interactions amongst the planets. This means the instantaneous (osculating) orbital angular momentum vector isn't quite constant, both in magnitude and direction. There are a number of different mean orbital elements that smooth out most of those tiny wobbles. One of those mean orbital elements sets (I'm not sure which one exactly) is used to determine the mean orbital angular momentum vector.

Planets don't quite rotate nice and smoothly because planets aren't perfect spheres and aren't rigid bodies. This gives the Moon, the Sun, and other planets a handle by which they can exert tiny little torques in the planet in question. The response of the planet to these torques is somewhat arbitrarily divided into two categories based on frequency. Very slow responses are called precession; faster responses, nutation. The non spherical nature of a planet means the planet undergoes a small torque-free nutation as well as the torque-induced precession, nutation, and polar motion. In addition to these, there are a number of terms (all small) that don't yet have a very good model behind them. This oddball terms, along with the torque-free nutation, are collectively called polar motion. The motion is fairly smooth if one ignores nutation and polar motion, and the effects are all small.

Precession, while very slow, can be quite large in magnitude. The rotational angular velocity vector that is used in determining the axial tilt incorporates precession but smooths out nutation and polar motion.