RandallB-- Your take, I would suggest, differs from that presented by A. Pais in his wonderful biography of Einstein, "Subtle is the Lord...The Science and Life of Albert Einstein. Pais in his own words, and in quotes from Einstein uses the three terms quite interchangeably -- particularly, see pp.440-449 in the Chapter entitled Einstein's Response to the New Dynamics. There's a good discussion in this chapter on Einstein's troubled with Quantum X, and some of his debates with Bohr -- all of which is discussed elsewhere in the book.
I will grant you that historically there is the Old Quantum Theory of Bohr-Sommerfeld orbits, particulate photons and so on; and the New Quantum Theory of Heisenberg, Schrodinger, Born and Dirac. But the Old theory had lots of problems, so it's only of historical interest, one that is, I suspect in decline.
And, note QFT can easily be fit into what you call the "narrower subject area of Quantum mechanics" It's all about the choice of Hamiltonian, and the representation of states -- like Fock Space for systems with indefinite numbers of particles.; the formal dynamical properties of QFT are well described by traditional QM/QT/QP-- see Dirac's book, many of the early papers by Heisenberg, Pauli, Dirac, Weisskopf on QFT and QED written in the 1930s.
Regards,
Reilly Atkinson