To show this is equivalent, note that if there is a machine formula_1 that enumerates all accepted inputs, another machine that takes in a string can run formula_1 and accept if the string is enumerated. Conversely, if a machine formula_3 accepts when an input is in a language, another machine can enumerate all strings in the language by interleaving simulations of formula_3 on every input and outputting strings that are accepted (there is an order of execution that will eventually get to every execution step because there are countably many ordered pairs of inputs and steps).