Hi,
The Codec does supply power to the Touch12. The older codecs were not designed with the Touch12 in mind, for support reasons, they included the PoE adapter. In reality, every codec has always worked fine with the Touch12s that I have seen, without the PoE injector..
If it works fine without the PoE injector, feel free to use it like that. If there seem to be power issues with the Touch12, might want to put it on the injector before calling TAC.