What are these "vibrations" that he constantly refers to? Why are they given special importance? From a biologist (or cog scientist) perspective, what are they?

I've been doing this for a while now and reached the end that the author calls partial enlightenment. I dislike the term however. It sounds so new agey.

The only thing that has happened is that I now see myself as an inseparable part of the physical universe both on the level of belief and alief. This insight makes life's problems seem less acute as they cannot target a permanent self on the alief-level. I still get sad, feel pain and pleasure but I see them more as fantastic stories about reality made up by my brain than the actual reality of (physical) cause & effect. I can also trust my instincts and subconscious mind more as I no longer fear losing myself when I surrender to pure feelings and motions beyond verbal thought.

This insight is only about a week old and I'm still making sense out of it so I'll end up with a more detailed description of what happened:

I progressed through the first two stages using other meditations but I quit smack in the middle of stage three. Back then I had no idea that feeling awful was a normal part of progress and I spent the next two years depressed in a Dark Night. Please don't do this to yourself! It sucks big time.

Last summer I ran into DavidM's article and started practicing. I worked through the pains of the Dark Night and finally broke out of stage three. However the boredom of early stage four sucked the motivation out of the practice and I took a break until I realized I was slipping back into depression.

So with the support of some friends and strong green tea I started meditating and hunting for the vibrations that permeate all experiences. I finally reached the end of stage four and could see the observer clearly. Like plucking out a flower I picked up the experience of the vibrating observer and realized that it was simply an automatic label the mind puts on thoughts and actions that it finds really important. Having now gained control of the label I could put it anywhere and immediately become it. So I became the universe and the mind was filled with joy.

The habitual mind (that is feeling like my usual self) eventually returned but it now has a dreamlike character and with a little meditation I can attach the feeling of self to basically anything and feel that I become that thing. I can also not attach the self to anything and just let the mind rest in a non-dual awareness.

After concentrating a while on my abdomen a painful wave grips my attention and physically throws me off my meditation posture.

Try to observe that carefully every time it happens. You said that you can sometimes see negative feelings as mental objects and not "yours," so you're definitely on the right track. Consider ways that you may not be fully seeing them as mental objects. For example, are the negative feelings afflictions for you? How do you know? If you know because they feel as if they are, make sure you recognize the feeling "being afflicted by negative feelings" or the impression "having negative feelings present in my experiential field" as a mental object too. (This "affliction" is part of the non-standard meaning of "aversion" or "hatred" which is shared with Buddhism.) It's a little hard to get across what I mean here, but if you see it, you'll probably recognize what I'm trying to say.

The vibrations I've picked up so far have been physical twitching of the abdomen, an in and out fading of the sensation of the abdomen, the abdomen themselves phasing in and out of existence and finally my attention itself setting and resetting itself. None of these experiences are very clear so I might be making them up.

Apart from the twitching, it sounds like you can perceive vibrations pretty well.

If you're in stage three, the "lack of clarity" is related to the fact that the "fading out" is is the clearest part. (Imagine a video of the moon waxing and waning using time-lapse photography, and imagine that the waning is extremely clear and the waxing is hard to discern.) That's simply how they present, and you may not actually be missing anything, which would mean you're doing a good job.

If you additionally feel that the vibrations aren't clear, that feeling is a different mental object; label it. Feeling that they aren't clear, and simple fact of their being unclear (or reporting on their unclarity using the word 'feel'), are different.

My concentration is still a mess but I'll try to find out the exact qualities of these experiences.

If you additionally feel that your concentration is a mess, that's a mental object too.

It is very hard to improve concentration in stage three, so if that's where you are, for now I wouldn't focus too much on trying to get a lot better at it than you already are.

Looking forward to hearing how it's going.

If you really are in stage 3, I would suggest not trying to shut out or ignore your negative feelings, whether or not you're focusing on your breath. Where are they, subjectively? What is negative about experiencing them? What are their exact qualities? Entertaining that sort of stuff can sometimes be helpful.

EDIT: My working theory right now is that the perception of "vibrations" is somehow related to the particular technique I describe, whereas the stages are more general in relevance.

Now this is a clear program with well defined goals and a clear roadmap to that goal. I'll definately be trying this out! I practiced a different kind of buddhist meditation a few years back. Unfortunately I quit right in the middle of stage three depressions. That sucked a lot and still sucks right to this day. Don't do that. I have no idea what you're talking about with these vibrations, though. The practice I did had a different focus. At the time of writing I'm only 45 minutes in but I'm already getting waves of negative feelings while trying to concentrate on my breathing in despair. I just hope I'll have a better resolve this time around to get over the bad stuff as quickly as possible...