How do pitch attitude, airspeed and sink rate come into play in a rapid descent or a dive?
Airspeed control is the difference. In a rapid descent, airspeed is controlled and the pilots will keep it below $v_D$, the maximum dive speed for which the aircraft is certified. For a modern airliner, this is somewhere around Mach 0.9.

In a dive with "nose down as far as they care to" (I read this as 90°), airspeed is not controlled, but an outcome of the situation. The airliner will go supersonic. It is possible to recover it from a short dive, but will incur damage to the aircraft (e.g. the pilots will need to lower the landing gear to increase drag, but lose the gear doors in the process).

If the pilots would only care to put the nose down to the point where they not exceed $v_D$, we are back at a rapid descent. They could lower the gear in order to increase the sink speed, and circle to increase drag further. Circling needs more lift (proportional to the inverse of the cosine of the bank angle), so more lift-induced drag is produced. To maximize drag, the pilots should fly at the maximum permissible load factor (which is rather low at maximum dive speed due to gust loads).

If we assume the L/D of the airliner to be 5 with gear and spoilers deployed, the pitch attitude would be between -11° and -12° and the sink speed would be between 50 and 60 m/s (10,000 to 12,000 ft/min). The bank angle would be between 45° and 60° and the plane would be violently shaken by buffeting.

Going any steeper than that would only be attempted by pilots with suicidal leanings. Flying like described above is scary enough already.