With a tube power amp, supposedly the only difference between the 4-Ohm tap vs. the 8-Ohm tap is that the 4-Ohm tap has more active secondary windings in the output transformer. Someone with more technical knowledge could maybe clarify this.
I don't doubt anybody's experiential findings, but to really get to the bottom of whether amp A does better in 4-Ohm mode or 8-Ohm mode, the SAME amp should be used for tests in both 4- and 8-Ohm modes. Amp A in 4-Ohm might fall on its nose compared to amp B in 8-Ohm mode, but too many other factors are involved to determine whether output impedance is at the root of the perceived differences.
Most solid state amps do not have output transformers (exceptions are many of the MacIntosh amps with their "autoformers") and have very low output impedances anyway. So, impedance output is basically a moot issue in most SS amps.
I would guess that some other factor is going on with the cj Premier 12's vs. the cj 11xs. In theory, all that's going on with lower impedance taps is more transformer windings in play. This usually translates to better control of difficult impedance load speakers, but perhaps a loss of transparency (maybe some additional phase shift--which may or may not be audible). With higher impedance load speakers, a lower impedance amp tap should not really affect its ability to drive the speaker per se.
I could be completely off on this. Maybe someone with more technical knowledge and experience in the area of transformers and impedance matching could maybe explain some of this stuff better.
In my system, the Super Eclipse IIIs do better on lower output impedance settings at the amp and sound better with more power than with less. They simply eat up either larger tube- or SS watts with gusto. Their bass extension and control with the 510 watt Musical Fidelity kW-500 amp, for example, completely revamped my impressions of the Supers power and operating preferences.