I think Wikipedia has a fairly good definition for this:
A stationary state is called stationary because a particle remains in the same state as time elapses, in every observable way.
What this means is that every observable quantity which can be computed from the state is constant in time.
In classical mechanics, this is kind of trivial because the state is just given by the positions and velocities (or momenta) of all particles in the system. In other words, the state is itself an observable quantity, and therefore, as you said, a stationary state is necessarily constant. But it's not particularly common to use the term "stationary state" in classical mechanics.
Where it really becomes useful is in quantum mechanics. Here, the state is more than just position and momentum. A quantum state contains some non-observable information, and that information can change over time even if all the observable quantities are constant. So in quantum mechanics, "stationary state" does not mean that all $t$ derivatives of the state are zero. The correct mathematical criterion is that these states are eigenstates of the Hamiltonian, $H\lvert\psi\rangle \propto \lvert\psi\rangle$.