A strongly driven (Omega much greater than gamma) two-level atom relaxes towards an equilibrium state rho which is almost completely mixed. One interpretation of this state is that it represents an ensemble average, and that an individual atom is at any time in one of the eigenstates of rho. The theory of Teich and Mahler [Phys. Rev. A 45, 3300 (1992)] makes this interpretation concrete, with an individual atom jumping stochastically between the two eigenstates when a photon is emitted. The dressed-atom theory is also supposed to describe the quantum jumps of an individual atom due to photoemissions. But the two pictures are contradictory because the dressed states of the atom are almost orthogonal to the eigenstates of rho. In this paper we investigate three ways of measuring the field radiated by the atom, which attempt to reproduce the simple quantum jump dynamics of the dressed state or Teich and Mahler models. These are spectral detection (using optical filters), two-state jumps (using adaptive homodyne detection), and orthogonal jumps (another adaptive homodyne scheme). We find that the three schemes closely mimic the jumps of the dressed-state model, with errors of order 3/4(gamma/Omega)(2/3), 1/4(gamma/Omega)(2), and 3/4(gamma/Omega)(2), respectively. The significance of this result to the program of environmentally induced superselection is discus sed. [S1050-2947(99)04809-X].