Mostly correct... but you have a few of the details backwards.
Yes, if you simply short the terminals of any dynamic speaker, you have the equivalent of connecting it to an amplifier with a very high damping factor.
And, if you simply leave the terminals unconnected, you have done the equivalent of connecting that speaker to an amplifier with a very low damping factor.
(Always remember that a high output impedance equals a low damping factor and vice versa.)
And, if you tap on the speaker cone, you will hear a difference that will let you hear how much that particular speaker is affected by amplifier damping factor.
And, yes, if you replace your paper clip with a transistor, and apply enough current to the base to turn that transistor fully on, the resistance across those terminals will be pretty much equal to RdON.
(It's actually a bit more complicated because you generally have two transistors, and the current involved is AC current, but the math works out pretty much the same.)
BUT your assertion about "the remainder of the current going back into and absorbed by the speaker" is not exactly correct.
Back EMF is a voltage...
The output stage of the amplifier is closing the circuit and providing a path for current to flow...
And, the lower the impedance of that circuit path, the more current that will flow, and the more "braking force" on the speaker movement that will result.
A lot of the energy not absorbed by the amplifier in this way is sort of trapped in the speaker in the form of mechanical energy...
Which is why speakers designed to work well with this sort of amplifier have plenty of mechanical damping to absorb that energy...
And why, when used with amplifiers without much damping, speakers that also lack mechanical damping tend to sound "sloppy and boomy"...
The speaker has energy stored in the form of momentum...
And, without anything to absorb that energy electrically through damping, it simply causes the cone to continue to bounce back and forth and generate sound after the signal driving it has stopped.
And, depending on the mass of the cone, that movement may continue for some time before the energy completely bleeds off.
It's the same reason why the speaker sounds boomy when you tap it without its being connected to anything...
When you tap it you are dumping energy into the cone...
And, without any sort of damping to dissipate that energy, the speaker continues to bounce back and forth until that energy is used up (either as heat or sound)...
However you are also neglecting another especially important factor... feedback.
In an amplifier without feedback the minimum output impedance is indeed pretty much the same as the on resistance of the output devices.
But most modern amplifiers have feedback applied around the entire amplifier.
And, when feedback is applied, the "apparent output impedance" is reduced by a MULTIPLE equal to the amount of feedback being applied.
Not only is the distortion reduced by a similar factor but the apparent output impedance is ALSO reduced by a similar factor.
Rather than simply providing a passive current path for current due to back EMF, the output device is part of a servo loop, and is
ACTIVELY sinking the current from the speaker.
This is how an amplifier can have a damping factor that equates to an output impedance FAR lower than the on resistance of its output devices.
If an amplifier has an open loop gain of 2000x (which would be unexceptional for a solid state amplifier)...
But its "overall gain"" has been set to 20x using feedback...
We have applied feedback that reduces the gain by a factor of 100x (we usually specify feedback in dB).
And, as well as reducing the gain by a factor of 100x, that feedback will also reduce the apparent output impedance by the same factor of 100x.
(So the output impedance, as seen by the speaker, will now be 1/100 of the on resistance of those output devices.)
Rather than "pulling the back EMF to ground by shorting the current it generates" we are now actively
DRIVING it to ground because that point is inside our "servo loop".
This is all part of "the standard theory of operational amplifiers"...
And virtually all modern solid state amplifiers are really just pretty much giant op-amps...
Your final summary is spot on... although opinions vary as to whether the time when tube amps ceased being current technology was "relatively recently" or "in ancient times"...
Speakers designed to work well with tube amps tend to sound "overdamped" or "dry" when used with modern amplifiers with a high damping factor...
And speakers designed to work well with modern amplifiers, with relatively high damping factors, tend to sound "sloppy" or "tubby" when used with amplifiers that have relatively low damping factors.
Incidentally... for those not familiar with the numbers here...
A typical tube amplifier will have a damping factor of around 4 or 5 (with 8 being considered quiet high)...
Whereas most solid state amplifiers have a damping factor of 500 or more (with 100 being considered quite low)...
(In practice any number much over 50 or 100 really doesn't make much difference.)
What you're talking about is NOT exactly wattage.
The ability to control the speaker is a matter of damping...
An amplifier with a higher damping factor will control the speaker more precisely... (within limits).
A dynamic loudspeaker acts as both a motor and a generator.
When you feed power into the voice coil it generates a magnetic field which pushes against the speaker's magnet's field and causes the cone to move.
Then, when you stop applying power, momentum causes the come to want to continue moving, so it acts as a generator.
The amplifier must then "absorb" the back EMF - thus acting as a "magnetic brake" that forces the speaker to stop.
Older speakers, intended to work with tube amplifiers, which have very little damping, had more internal mechanical damping, and so relied less on the amplifier to provide damping.
Modern speakers, designed to work with amplifiers that have a lot of electrical damping, tend to have less mechanical damping, because they expect the amplifier to provide the damping required to control the driver.
The ability to absorb back EMF does vary between amplifiers... and more powerful amplifiers... and amplifiers with a higher damping factor... tend to be able to do it better.
Many amplifiers that can deliver large amounts of output current also handle this better than some with limited current capabilities - but the relationship is not totally simple.
Also note that this is not at all an issue with electrostatic panels - which do not generate back EMF via this mechanism.
Actually? You can TEST this with nearly any dynamic speaker and a PAPER CLIP.
First? Remove the speaker wires from the back of the speaker. Than? Around 'front', go ahead and 'flick' the edge of the river with a finger snap. It'll make a certain noise. Put that noise in Short Term Memory.
Second? Short the speaker terminals WITH the Paper Clip. Go back around front and REFLICK. Very much different noise.....
If you look at transistor specs? You will see something called RdON. Resistance of the device when ON. This is a VERY low number. This resistance will be related to HOW MUCH of the BACK EMF the amplifier 'absorbs' and tuns into HEAT. The remainder? Basicall goes BACK to the speaker in the form of a SHORT which instantly consumes the energy.......This is HIGH DAMPING.
Now? this is historically fairly a recent development. Tube amps have a VERY high resistance when compared to Transistors. And the speakers? Had a LOT MORE self-damping. This is the source of some BASS PROBLEMS where you run a certain speaker and either the bass is 'Tubby' or 'Missing In Action'.....depending of if you are running a speaker which would PREFER a tube or Transistor amp on 'the other one'.......