Adiabatic means quasi-static and isoentropic - slow enough to create negligible amount of irreversible excitation. This is the common rationale of technically different definitions. E.g., Landau & Lifshic'es definition has two components - thermally isolated (to prevent entropy change by heat exchange) and slow (to prevent irreversible excitation). For a gaped quantum system adiabatic can by quite fast (just keep Planck constant times the characteristic driving rate below the value of the energy gap). 

What is confusing indeed is that there can be an intermediate speed which you can be reasonably adiabatic - much faster than heat exchange with what you separate as the "reservior" but much slower than the equilibraton speed of the degrees of freedom being excited. That's why adiabatic can be fast and slow at the same time - there are two conditions to satisfy. These subtleties are often not made sufficiently clear,  
but that's what we have physics.SE for :) 

To sum up, don't make "waves" (entropy) and you'll adiabatic.