Using a 24 volt ac relay with a 16 volt ac doorbell circuit

As far as I know, all the switching for these is done at the control box. Rather than a centrifugal switch, I believe the relay is simply a fixed time-delay contact.

But I consider them archaic these days - For single-phase supply, I buy 2-wire pumps, and would replace any 3 wire pump with a 2-wire pump when the 3-wire pump stopped working. The arguments in favor of 3-wire have become much like arguments in favor of a buggy-whip to make your automobile go faster...

If 3-phase supply is available that's a whole different animal, but it's uncommon for residential applications where I live.

• 1/2 hp is a motor rating. Motors have a whole bunch of inductive kick, so controlling relays must be de-rated.
• 20A is the rating at 277V. Interesting thing, 277V was commonly used in industrial lighting, where they were interested in a high ratio of lumens to copper (40kw on twelve 12AWG wires); so 277V almost implied HID lighting with magnetic ballasts (sodium, metal halide, etc.) This is called a "ballast load" because they too have an inductive kick, though not like a motor. This seems to imply a 20A rating for ballast load.
• 20A is the rating at 120V. 120V lighting tended to be incandescents, i.e. a simple resistive load with no kick. Incandescents do have an inrush current which is considerable, so contacts need to be a little tougher than a pure resistive load. By the way, electronic ballasts and LEDs also have inrush current from the capacitors/inductors charging up.

The 30V DC is approximately what you get when you take a common 24VAC thermostat or doorbell transformer and rectify it to DC. Very appropriate for a 1960s system.