I'd like to mention that:
the circuit CMMR (caused by resistor mismatch and OpAmp CMMR) is not the main issue here. It does affect the "diamond shape", as do the input voltage offsets, finite gain, output current and temperature drift. The "diamond shape" seems to be extracted only from the input and output limitations (including the internal nodes) when compared to the power rails. CMMRs affect the whole range linearly (as others non-ideal characteristics) and what the shape shows is the limits for linear operation.
splitting the gain resistor is very interesting to break the problem down to simpler independent ones, but the analysis works for ideal OpAmps and only when the internal voltages don't matter. The "central tap" changes the way the currents flow in the input stage resistors and pulls the "middle point" of the component to ground, affecting the common mode voltage at the inputs and the absolute output voltages of the first stage. In fact, as mentioned in another answer, it is great technique to improve the control over internal common mode voltages.
many INAs have unity gain in the second stage. In these cases the gain is not divided in two stages, but "moved" to the entry stage. That is a problem regarding common mode voltage. The absolute voltages in the internal nodes depend on the input common mode voltage, even with gain = 1 (which transfers the CM voltage to the internal outputs, like in the second circuit shown in the question).
Inputs and outputs affected by common mode:
Regarding the specific question, the "buffered differential amplifier" is a special case for the full INA, where Rg = infinity. It can even be simulated with the diamond tool, from analog devices, for INAs with unity gain in the second stage.
The diamond is generated from the intersection of 3 polygons, which generate an area which is compared to the ideal operating region (which is a rectangle). Graphically, is works like this:
The amplifier will work linearly if the entire input range (Vdiff and Vcm) generates an ideal output (which also depends on Vref) which fits completely on the intersection of the 3 polygons. The ideal output is shown as a red rectangle.
The blue parallelogram refers to the output limits. It follows the voltage rails, according to the component characteristics, and is shifted up and down with Vref.
The green rhombus refers to "internal nodes" and the information needed to draw it is usually not available in the datasheets. The highest and lowest points are centered on Vref and the shape varies with the voltage rails. In all components I simulated, the shape is not affected by gain (and this is strange).
The orange rhombus is related to input limits. The highest and lowest points are also centered on Vref and the shape varies with the voltage rails and gain. The larger the gain, the wider the shape. Since the two rhombuses are very similar for gain = 1 in all INAs I checked, increasing the gain makes the "diamond shape" less dependent on the orange rhombus.
You reached here!? Well, a direct answer to the question is: it depends, and for real integrated INAs, it does not make a huge difference.
If you go beyond the question and really want to solve CM voltage issues, go with specific solutions like the one mentioned for ECG.