That is a more appealing definition for sure. It resonates with what most of us intuitively feel about consciousness. However, it is not a workable definition because it doesn't describe the "process".
That's actually not accurate.
Remember, there was a time when the northern lights were just "those colored lights you can see in the sky sometimes near the north pole".
There was no description of the process that caused the lights, but still it was a perfectly fine description so that folks could identify the lights and study them.
And the current definition of consciousness works just fine for the current science.
In fact, it's still perfectly workable even though we do indeed have a much more specific definition than we used to.
We can describe the various brain waves associated with different states of consciousness, and thanks to the very recent deep brain studies we can describe the behavior of the "signature waves" which indicate the presence and absence of consciousness.
We can describe various states of consciousness in terms of the 3 neural axes I described earlier (activity, focus, concentration) and their corresponding experiential qualities.
We can identify brainwide neural behavior associated with consciousness, such as the patterns of syncing of neural pulses during awareness and during absence seizures.
So we have a workable high-level description which allows us to study consciousness, and groups of indicators which allow us to determine when it is present and what mode it's in... in fact, this knowledge is now allowing us to determine the conscious states of "locked in" patients... but not yet any theory which connects the experience and the brain activity in a clearly explanatory way.
But we're making progress with the definition we have... which appears to be perfectly acceptable to hard-nosed neuroscientists, but problematic to philosophers with time on their hands.