the confusion may be because there are actually two output impedances from an amp - the static output impedance and the output source impedance. they are not the same thing.
an amp with a zero output source impedance can still have a 120 ohm output impedance. this is because you are dealing with AC signals (sound) rather than DC signals. it's like saying that a zero source impedance amp is made to drive 8 ohm speakers. put a 2 ohm speaker on it and the transistors will fry because they cannot handle a near short and still deliver current and voltage.
on my Crown D60 amp, the output impedance is 15 milliohms in series with 3 microhenries. it cannot drive a 1 ohm speaker. it has .01 ohms Zo from DC to 200 Hz, .02 at 1 KHz, .16 at 10 KHz, and .3 at 20 KHz.
As far as damping factor, it is 800 from DC to 100 Hz. But the nominal output phase angle shifts from 0 degrees at 20 Hz to 80 degrees at 10 KHz.
huh? well, bass seems to radiate from everywhere. if it is just heard as coming from in front of you, the triangles would seem to come from the right (90 degrees). since the bass seems to come from everywhere, if you could phase shift it so that it comes from the left, then the triangle would seem to come from in front of you. this is a over simplification. phase shifting, as seen on an oscuilloscope would be on the z-axis, and therefore represents 3D spacial information. or what some can think of as soundstage depth and / or height (depending on the frequency).
now as far as damping factor - some say that it doesn't really matter with headphones, others seem to think that it does. it will all depend on what part of the sound spectrum is being listened to, and the resonance frequency of the driver. damping factor is applied across the whole spectrum, whereas point of resonance is strictly at one point (the AKG K501 being an exception).
if you are to "look" at an amp, you must take into consideration nominal phase response along with phase shift due to capacitance , resistance and inductance (reactance). one of the most important things to look at is the slew rate. the higher the slew rate the better because signals will tend not to blurr or mush. there will be no "over-ringing".
headphones are reactive devices, they contain coils which have inductance (millihenries) and resistance (milliohms), and as such the inductance interacting with the the capacitance of the cable will cause all sorts of phase shifts in an amp. beyond a certain range the amp will go from being capacitance driven to being inductance driven (or is it the other way around?). the point at where the shft is made may be discernable enough to make you think that the sound becomes recessed. or you just may perceive it as a collapsed soundstage. as it is, the higher the frequency, the less the channel separation which the amp can produce. which is why separate channels with separate power supplies are best. of course you'd have to work your way all the way back to the source, and I do not know of any CD or DVD player with twin beams working at the same frequency reading the same bit at the same time. it would be like playing two mono CDs at the same time (time sync) so that the sound is pristine. for all intents and purposes it cannot be done. but it can be engineered real close.
if you take a look at your typical opamp, you should see at least two resistors on it's output, with one possibily being used for feedback. these two resistances will "interfere" / "interact" with the output resistance seen (headphones).
as far as the headphone impedance being higher than the amp output impedance, this is so that there is not a "motor" generated. it's like the output impedance of your DVD player being 600 ohms, but the input impedance of the pre-amp or amp (next stage) being 5,000, 10,000 or 25,000 or 100,000 ohms. typically it sound be 10x. the next input stage is "loading" the previous stage. if it were otherwise the current flow would be from the amp to the dvd (or from the headphone to the amp). the amp would therefore burn up the dvd.
there is a "nominal" headphone output impedance, typically measured at 1 KHz. Other manufacturers will list what it is the lowest at at any frequency. they are just two different measurement standards.
i believe that most headphone amps just put a resistor in series with the output. as such it is increasing the resistance seen by the headphones (making them sound less bright), but they are also shifting the phase (even if it is so slight). Just "so slight" may be enough to shift a peak or a dip out of a certain range (make it less harsh). It all depends on where that dip or peak occurs. if it is a peak in the upper midrange, it could shift it out of the vocal range. if it is in the bass region it could become more or less boomy.
now, most replacement cables actually LOWER the capacitance. this lowering will also produce a phase shift. it should sound a little cleaner, with the bass a little tighter. this is because the overall reactance is reduced, and with it a phase shift occurs later or further up the sound spectrum. the "seen" signal is cleaner, there is less over-shoot because the amp is not fighting stored energy (capitance / capacitor effect). and just as there is capacitance from one channel to ground, there is also capacitance between the capacitances (or from one signal wire to another signal wire).
the reason why Grados sound the way they do is because they are basically electrosets or pizo tweeters. they are (I believe) built differently than regular coils over core bobbins. i think that they did a great engineering job.
so, it is not because of output impedance that a great amp sounds great - it is because of a high slew rate, the inherent sonic quality of the transistors / tubes / op amps, and because of the nominal phase shifting.
you have to try every amp with every headphone with every source and find one that is pleasing to you.
other than tubes, most of today's amps have less than 1 ohm nominal (source) output impedance. what decides if they sound good is the resistor network on the outputs. rather than just adding a 120 ohm resistor on the output resistor network, they should be reconfigured so that their interaction presents a different output impednace while still keeping the feedback within tolerance and the output voltage within acceptable levels.
just adding a resistor will cut down on brightness, and therefore the overall sound quality is "shifted" downward whereby it may sound boomy. the opposite is also true - put a low impedance headphone on a high output impedance and the sound could get really muddy or bright (no bass).
I'm sorry if this all sounds confusing. there are much better explanations in the amp section. just do a search for "impedance," "reactance," "resistance," or "inductance".
it's all in the engineering. different tubes, transistors and opamps sound different because they are made differently. in the case of tubes, it is the grid area and the gas content. in the case of opamps and transistors it is the gate and substrate materials. the purer the material, the more likely it will sound better.