Do I want a common trip or independent trip quad breaker?
Addressing your points in turn:

1) Handle ties do not make a BR breaker pair common trip (2-pole BR breakers have an internal common trip mechanism) -- see footnote 2 on page 84 of Eaton's catalog.

2) Common trips are generally used in 240VAC circuits -- while what the previous installer did with the pump circuit is not wrong as per the 2014 NEC 240.15(B)(2):


  Grounded Single-Phase Alternating-Current Circuits. In grounded systems, 
  individual single-pole circuit breakers rated 120/240 volts ac, with 
  identified handle ties, shall be permitted as the protection for each 
  ungrounded conductor for line-to-line connected loads for single-phase
  circuits.


common trips are generally used on all 240VAC circuits (it's hard to get a non-common-trip 240VAC breaker in a typical form factor), and are needed in mixed 120/240VAC circuits for the reason you observed, as per NEC 240.15(B):


  Circuit Breaker as Overcurrent Device. Circuit breakers shall open all 
  ungrounded conductors of the circuit both manually and automatically unless 
  otherwise permitted in 240.15(B)(1), (8)(2), (B)(3), and (B)(4).


(The exemption in (B)(1) is for multiwire branch circuits serving only 120VAC loads, while (B)(3) and (B)(4) are irrelevant to residential practice.)

3) You are correct -- this is documented pictorially in the table headings on page 82 of the Eaton catalog, so it's easy to miss.