> feeding signal into the emitter. Your second circuit is the buffered version of that.
I don't see that.
It's hard to find examples one the web because they aren't a common pattern. However, if you start with this one,
From,
https://electriciantraining.tpub.com/14180/css/The-Two-Input-Single-Output-Difference-Amplifier-92.htm
R2 is just for biasing (it also lowers the input impedance of input 2).
Input 1 is high impedance and input 2 is low impedance. So the next step is to add a buffer to feed into port 2. At first think of the buffer as AC coupled - no problems there.
The next step is to remove the AC coupling and make the buffer DC coupled. If we choose a PNP transistor for the buffer we get your circuit but with the resistor between the emitter set to zero. If we choose an NPN transistor for the buffer we get the common differential amplifier.
There's one more detail. When we DC couple the input impedance into the top transistor is the same as the output impedance of the buffer stage. That means we loss some gain. However we don't lose the differencing. The difference voltage appears across the two 're' transistor emitter impedances. When we add a resistor between the emitters we further reduce the gain. Adding the resistance between the emitters might seem weird but in fact we do it all the time when we add RE resistors to a diff-pair to reduce the gain, as done in many amplifiers.
The important difference between these variants and the common differential amplifier is the common differential amplifier is balanced for DC and will do differencing on both DC and AC signal. The variants do differencing on AC, sort of DC differencing as well but there's DC offsets that we need to factor in. You can see an example of that here, the reference voltage is the zener voltage less the two Vbe drops,

In full your circuit is like an unfolded differential pair. It's actually got a name but I can't remember it right now. The folding and unfolding ideas are common in oscilloscope and wide-band output amplifiers for signal generators. The books by Feucht might have the details.