As Soundhound explained (from what I remember), the standard is whether playback is indistinguishable from live. The only real way to test this, though, is to have a live performer standing in your room between your speakers. You record (with a mic that's known to be flat in its response) that performer in that position, then switch between the performer playing live and playback through the speakers. If it sounds the same, then it's hi-fi.
While that is correct in principle, the execution of the test is a bit more involved. If the test was carried out in that manner, the sound that arrived at your ears during the recorded portion would have a double dose of the room’s acoustic signature (reverb etc picked up by the recording microphone, and again through the speakers), while the live portion would only have a single imprint of the room’s signature.
The way these tests have always been done is to record the music under anechoic conditions, either in an anechoic chamber or outdoors, away from reflective surfaces. When the recording is then played in the listening room, it is receiving a single dose of the room’s acoustic – the same acoustic which is influencing the sound of the live musicians.
In the 1950s and 60s these “live verses recorded” tests were extremely popular marketing tools for companies like Acoustic Research, Wharfedale and others. I have a photo of one such event being carried out by Wharfedale in Carnegie Hall. By all accounts most of the audience was completely unable to tell the recording from the live musicians.
This has always been the classic test, and the original intent of a High Fidelity reproduction system. That system includes everything from the microphones, through the storage device (disc or tape in the earlier days), and on into the home.
Of course musical tastes have changed since those days, but it should be noted that even in the 1930s, since the development of electrical recording, multiple microphones and rudimentary mixing and equalization controls have been used, at least in popular music recordings. Classical music recording has always hued more to the literal definition of High Fidelity, and this continues today, although artistic license is and has always been taken with shaping the original sound to give more of a sense of “subjective reality” to the listener at home – but that is another long and complex topic for recording engineer geeks.
With modern popular music, and in fact going all the way back to Les Paul's invention of modern multi-track recording, there is no “original sound” which can be reproduced since processing has taken place either through such simple techniques as using non-flat microphones to complement a particular instrument or singer (in the early days, that and rudimentary EQ was all that was available) to more drastic solutions today using DSP. Therefore the term “High Fidelity” cannot be literally applied to this type of music / recording technique, no matter what the home system sounds like.
However that does not change the core definition and original intent of High Fidelity; accuracy to the live event, any more than the concept of accurate color rendition in photography and video to the original can be changed, and in the case of High Fidelity, that principle continues to be best demonstrated through the classic live verses recorded test.