Modeling channel response as complex vs splitting I and Q
After giving it more thought, I see my original confusion. We need to be careful about what we mean by the channel's impulse response. Normally for an equalizer for this application, we are not modeling the channel alone, but rather also including the frequency mixing up and down, before and after, transmission through the channel. When we include the mixers in this way, we get the cross coupling as I mentioned for the first case. So if $h(t)$ is the response of the channel alone, then $g(t)=e^{-j\omega_ct}h(t)$ is the response of the entire system from baseband to baseband. The second case with independent responses would be for applications where the channels are not mixed together at any stage and are transmitted and received independently.