Let me try to clarify, since "balanced" is a term that seems to be losing its meaning in widespread usage.
"Balanced" refers to a type of interconnect with two signal wires, and the signal is the voltage difference between them - independent of any signal common to both wires. The best performance is achieved when the impedance to ground is the same from either line, and the impedance to ground is small at the source but large at the destination. It has nothing at all to do with how the signal is amplified, and there is no requirement that the signals be equal in amplitude relative to ground; in fact the whole point it to make them independent of ground - it's only the difference between the two wires that matters. There is no such thing as a balanced amplifier; interconnects can be balanced or unbalanced.
"Push-pull" is the term that means there are two equal and opposite signals relative to ground. An amplifier with push-pull inputs and push-pull outputs can accept balanced inputs and produce balanced outputs, but it is easily overloaded by a common-mode signal at the input. However, this can be done without expensive transformers, and many so-called "balanced" inputs and/or outputs are achieved this way. This cost-cutting practice has fed the confusion of terms. There are no push-pull interconnects; amplifiers can be single-ended or push-pull.
The final bit of confusion comes with headphones. Most headphones and headphone jacks have three wires for stereo - two unbalanced signals and one shared ground. In many cases, you can get a sonic benefit by using four wires, so the return signal current is not shared on a single ground wire for both channels. Such a headphone cable is a balanced cable, since there are two wires for each channel and the signal is the voltage difference between them. This is true even if one side of each pair is grounded at the amplifier. The impedance of the amplifier output to ground is of course not equal to the impedance of ground to ground (i.e. zero) so it is not an optimal balanced interconnect application, but the headphone does not much care, it only sees the difference between the two wires and knows nothing of any ground potential. You can achieve a more optimal source by using a transformer with a center tap (like Smack) or a push-pull topology or simply adding a resistor to ground from the second interconnect wire.
There are many more things that can be said on the subject, but this post has become pretty long already. Bottom line, any headphone can be converted to balanced lines. Then you can wire one line of each side to the "shield" connection at the plug and use a normal headphone jack, or you can use a pair of XLR connectors, or some other connector that will accommodate four wires for stereo. In the latter two cases you can use an amplifier that has more-optimal impedance balance between the two wires.
The DAC claim makes no technical sense, so I can't address it. Might have bee written by the marketing department? :^)
