I think the two of you are talking past each other. Perhaps this will help: decoherence means that the "alive" and "dead" terms in the wave function, while they are still both there, cannot interfere with each other; in other words, there is no way to reverse the interaction that created the "alive" and "dead" terms in the wave function, the way that, for example, you can take the two terms of the wave function of a qubit that has just had its spin measured, and recombine them by reversing the spin measurement (putting the two output arms of one spin measurement device into the input arms of a second spin measurement device to recombine them so only one output arm of the second device will have a signal).
In other words, both of you are making true statements with the appropriate intepretations of the terms. The cat is in a superposition of "alive" and "dead" (an entangled superposition, in which the "alive" and "dead" states of the cat are entangled with the "not decayed" and "decayed" states of the radioactive atom), but there are no terms in the wave function for states that are linear combinations of "alive" and "dead" (corresponding to, for example, a spin-X up qubit's state written in the spin-Z basis).