I think you raise a very legitimate question. A lot of psychology experiments ascribe importance to electrical or fMRI activity associated with 'tasks'. Consciousness studies have tried to pin down 'signatures' of consciousness based on such activity patterns. However, we have every reason to think that a psychology subject taking part in an experiment is just as conscious of just as many things up to the point of responding to a task as during the task. There was a very neat paper presented by a post-doc from Cambridge at the 2015 Helsinki Consciousness meeting which made the point. If subjects were asked to respond to letters and were shown letters they showed a change in brain activity from 'resting' to 'conscious of stimulus' when a letter was shown for long enough. But if they were shown a number there was no change despite the fact that they could say afterwards that they had seen the number and deliberately ignored it.

The literature is complex and you may find it interesting to read the books by both Dehaene and Northoff on the neural basis of consciousness. However, my impression is that there are unjustified assumptions involved, as you intimate.

My suspicion is that many of the signals that psychologists hope are signs of 'task versus rest' are merely indications of 'camera shot changing'. If you play your holiday snaps on iPhoto you can get the snaps to drift gently back or forward, separated by a 'shattered glass' transition. I strongly suspect that if you analysed the image for 'activity' in the sense of maximal change the shattered glass transitions would show up like a sore thumb. But they are of no real importance to the slide show. If processing information about a particular set of input data requires complex co-ordination through domain specific synchrony then you would expect a transition to a new state to show up as a wild fluctuation in potentials as new synchrony patterns were set up. When a train goes by you do not here the wheels turn on the rails, just the clack of moving from one rail to the next. My belief is that neurobiology is still at the stage where it cannot determine whether signals are 'data' or transitional mechanics.

One or two neurobiologists have built very specific theories on  the 'resting state' concept. I am sceptical, as you can see. Hard science has feared to tread in such areas.

I think I could help you only by giving some informations about Georg Northoff's resting state studies.

According to Northoff, the resting state isn't a simple state of non-activity of the brain, indeed there is an intrinsic activity that could be important for the self-processing and the interaction with the extrinsic stimuli from the world.

There are some important correlation between self-processing and the "default mode network", the intrinsic activity of the brain during the RS: https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3872301/

Moreover, Northoff tries to compare the kantian view of the spatiotemporal structure of the consciousness with the default mode activity of the brain in RS: "the brain’s resting-state activity may structure and organise stimulus-induced activity in such a way that the latter can be associated with consciousness, self, and spatiotemporal continuity": https://www.ncbi.nlm.nih.gov/pubmed/22748399

As for the conscious states, the so-called neural correlates of the "actual" consciousness are not sufficient (only necessary) to explain the phenonemon. Northoff describes also what he calls neural predisposition of consciousness, that is the necessary non sufficient conditions for a "potential" consciousness, in other words the intrinsic structure of an developing state of consciousness. This neural predisposition is trackable in the intrinsic activity of the brain, without any particular cognitive task involved:

https://www.ncbi.nlm.nih.gov/pubmed/25669313

https://www.ncbi.nlm.nih.gov/pubmed/26290126

https://www.ncbi.nlm.nih.gov/pubmed/23253946

You have clearly studied Northoff's writing in detail, Federico. I find it intriguing but also find it hard to know what it could mean in physical science terms. Do you have any conception of what 

"the brain’s resting-state activity may structure and organise stimulus-induced activity in such a way that the latter can be associated with consciousness, self, and spatiotemporal continuity":

might mean? I end up thinking it could mean anything or nothing!

It's a relief to see that I wan not the only one thinking that the abovementioned Northoff's statements were someway obscure!

Not that I'm blaming here: it looks like sheding light on the psychological meaning of the resting state network won't be an easy task...

(Other answers are welcome!)