A better way to look at it is that for a stiff problem, any stable explicit calculation leads to an error that is much smaller than the required error tolerance.
There are many good methods for automatically detecting stiffness using explicit schemes, especially embedded Runge-Kutta pairs. See for example:
- Detecting stiffness with the Fehlberg (4, 5) formulas
- Stiffness detection and estimation of dominant spectrum with explicit Runge-Kutta methods
- Stiffness Detection Strategy for Explicit Runge Kutta Methods
In faleichik's second example, as the step size is reduced, one would see a sudden dramatic decrease in the error to levels far below a typical desired tolerance as the stable timestep threshold is crossed. So a good error estimator would indeed reveal the stiffness of the problem. In the first problem, the error obtained with a stable step size would be in the range of the typical desired tolerance, indicating non-stiffness.
Note as a consequence that any problem becomes non-stiff if a sufficiently strict error tolerance is required.