"Authority" is a new one for me! But yeah, it means whatever you want it to mean... "Audiophiles" and reviewers use a lot of meaningless words.
I'd ignore anything other than clear language... If they say
"it goes loud", or the
"the bass is weak", or
"I can hear distortion", those words have real meanings.
There's output voltage (or power*) as well as
noise, distortion, and frequency response, and that's about it!
Distortion is almost always below audibility unless you drive the amp into clipping. If you have an integrated DAC/amp the amp usually has enough headroom to it's impossible to clip the analog amplifier... You'll only get clipping if the digital data is clipped (which can happen if you boost with digital EQ).
Frequency response is almost always flat except if the source impedance is too high (relative to the headphone impedance). The headphone
impedance isn't flat across the audio range and the impedance variations can result in frequency response variations. Most headphone amplifiers have low-enough impedance that this isn't a problem, and higher impedance headphones are more immune to this.
Noise can be an issue with some computer soundcards, and of course more-sensitive headphones make everything louder including the noise. Noise is most noticeable when there is no signal or when the signal is quiet. Often the noise doesn't go down when you turn-down the volume control, and any noise generated in the amplifier doesn't go down if you turn-down the digital volume so if you listen at low volumes the signal-to-noise ratio can be worse and sometimes distracting/annoying.
I don't have a dedicated headphone amplifier but my laptop only goes "moderately loud" (with any of my headphones). Plus I use ReplayGain with music which brings-down the volume on most songs. The headphone-output on my receiver goes plenty-loud.
* Power (wattage) depends on voltage and impedance (lower impedance means more current and more power). Then, sensitivity depends on how efficiently the electrical energy is converted to acoustic energy.