試飲 is a general term for sampling a drink, often in a commercial setting, without the connotation of expert evaluation that 味利き carries.
Etymology
Compound of 味 (aji, 'taste') and 利き (kiki, 'effectiveness, discernment'), from the verb 利く (kiku, 'to be effective, to function well'). The spelling 味聞き uses 聞く (kiku, 'to listen') metaphorically.