For light, GA aircraft, is there a "standard" turn, and what does that mean?
The FAA Aeronautical Information Manual defines 'standard rate turn' as,


  A turn of three degrees per second. 


This results in the completion of a full circle in 2 minutes. Every airplane must be able to complete a standard rate turn in order to be certified. The standard rate of turn is used as the main reference for bank angle.

GA aircraft are expected to turn at the standard rate of turn during holding, intercepting, tracking, approaches, departures etc. 

You're correct that the airspeed determines the turn rate. As the turn rate is standardized, the bank angle becomes dependent on the air speed. From FAA-H-8083-15A Instrument Flying Handbook,


  True airspeed determines the angle of bank necessary to maintain a standard-rate turn. A rule of thumb to determine the approximate angle of bank required for a standard-rate turn is to divide your airspeed by 10 and add one-half the result. For example, at 60 knots, approximately 9° of bank is required (60 ÷ 10 = 6 + 3 = 9); at 80 knots, approximately 12° of bank is needed for a standard-rate turn.