I don't think the first sense and the second are mutually exclusive.
A dog has half as much processing power as a human = a dog can think the same thoughts but only at half the speed.
A dog has half as much consciousness as a human = a dog is only half as aware as a human of what's going on.
And yes, I definitely think that this is how it works. For example, when I get up in the morning I am much less aware of what's going on than when I am fully awake. Sometimes I pause and go "wait what am I doing right now?" And of course there's those funny times when you end up going to your old school by accident because you're not aware of what you're doing until it's too late...
The only part I disagree with is "A being that is functionally exactly like you, and that is experiencing exactly what you are experiencing". A being that is only half as conscious is going to have a different brain, so will act differently. You definitely notice when someone is only at 50% consciousness.
Finally, I submit that consciousness works just like will power: It is a limited resource which you can allocate to certain tasks/thoughts, and by training it you can get more of it. A good rationalist needs a large pool of consciousness.
This paper, or more often the New Scientist's exposition of it is being discussed online and is rather topical here. In a nutshell, stimulating one small but central area of the brain reversibly rendered one epilepsia patient unconscious without disrupting wakefulness. Impressively, this phenomenon has apparently been hypothesized before, just never tested (because it's hard and usually unethical). A quote from the New Scientist article (emphasis mine):
One electrode was positioned next to the claustrum, an area that had never been stimulated before.
When the team zapped the area with high frequency electrical impulses, the woman lost consciousness. She stopped reading and stared blankly into space, she didn't respond to auditory or visual commands and her breathing slowed. As soon as the stimulation stopped, she immediately regained consciousness with no memory of the event. The same thing happened every time the area was stimulated during two days of experiments (Epilepsy and Behavior, doi.org/tgn).
To confirm that they were affecting the woman's consciousness rather than just her ability to speak or move, the team asked her to repeat the word "house" or snap her fingers before the stimulation began. If the stimulation was disrupting a brain region responsible for movement or language she would have stopped moving or talking almost immediately. Instead, she gradually spoke more quietly or moved less and less until she drifted into unconsciousness. Since there was no sign of epileptic brain activity during or after the stimulation, the team is sure that it wasn't a side effect of a seizure.
If confirmed, this hints at several interesting points. For example, a complex enough brain is not sufficient for consciousness, a sort-of command and control structure is required, as well, even if relatively small. A low-consciousness state of late-stage dementia sufferers might be due to the damage specifically to the claustrum area, not just the overall brain deterioration. The researchers speculates that stimulating the area in vegetative-state patients might help "push them out of this state". From an AI research perspective, understanding the difference between wakefulness and consciousness might be interesting, too.