I suggest that the seeming incoherence of moral intuitions that you pointed out and which arise from moral experiments such as the Trolly problem, may be less an indication of the differences between our true moral intuitions as much as an indication of how we are all caught in the dichotomy of being disinterested vs invested agents.
Part of what is characteristic of a moral dilemma in my opinion, is that it something that challenges the personal value system upon which we identify ourselves, and it is a scenario in which we have the ability to intervene. Under such a scenario, knowing to have the ability to intervene and choosing not to, would put us in a position where our inaction would, much as our action would have, serve to define us.
Consider now that the way we have been socially engineered or "socialized" has been under utilitarian inspired principals and the values of consequentialism - arguably necessary in the assessment of the rights of persons who constitute part of a collective.
Its not difficult to see that any examination of a moral dilemma will evoke two necessarily different reactions from us: First, a very personal and vested consideration of event elements and possible implications for the self as an agent of intervention / non-intervention; Second, a utilitarian based, necessarily dis-interested assessment of what would be the appropriate reaction as a non-invested agent in the scenario.
I suggest that the dilemmas such experiments flush out are really ones existing within each of us as an incoherence between social engineering and an intuitive moral response. We have been socialized to assess moral problems from the viewpoint of one who is identified primarily as part of a collective (personally detached), Simultaneously we are instinctively moved to make an assessment as an agent who is continuously defined by his / her choice of action / non action.
Then the response to the moral dilemma, in the end, is measured again against consequential implications in the context of the collective.

There are known, unknown, and unknowable consequences of my actions and inactions in every scenario. My default stance is non-interventionist. Pushing the fast man onto the tracks to save the cute little children assumes that the net benefit of the cost/benefit trade-off is positive (4 children > 1 fat man). However, we don't know what the value of the children's lives versus the value of the fat man is. For all we know the children will become a band of bandits who will terrorize society for years to come, whereas the fat man is a cancer researcher on the verge of a break-through.
If I know that my actions will improve the situation in positive directions all around (no cost in any direction) then I will act. If there are unknown or unknowable effects of my action then I must default to inaction and let circumstance take it's natural cause, as if I wasn't there... unless of course I have some other overriding responsibility or duty to act in a certain way that agrees with my morality.
I believe this stance prevents the trauma surgeon from harvesting a living person's various organs to save multiple accident victims.

I think we misjudge both the value of moral intuitions and the time frame when they appear and are of use to us. While the trolley car model appears to be a way to clarify our moral options in a particular situation , in fact moral intuitions normally appear after something has happened, not before. And most often they appear not in relation to our own actions but to those of others. I think our morality is applied retrospectively, after an action. Our moral intuitions and in general our moral senses are rarely prescriptive, instead they're "critical" in the aesthetic sense. And our moral intuitions are similar to the sorts of judgments we make about art works or drama or music. The role of the moralist like that of the art critic is to create a vocabulary to appreciate the different values informing human actions and to judge them in relation to context. Moral action like art almost always becomes more opaque and complicated as we examine it critically. The value of morality then is to explain human action to us, just as any story tells us something about the kinds of people we are and perhaps want to be. But it does not tell us the kind of people we 'ought" to be.

The problem with the "trolley problem" can be illustrated by stepping outside moral philosophy into jurisprudence. In law, the actions of an individual are sometimes compared to the hypothetical action of a fictional "reasonable person." When there is no reasonable action, as is the case in the trolley problem, any legal (or moral) argument seems moot.

What bothers me about this dilemma is that "You" actually have another choice that was not discussed: "you" could jump off the bridge yourself. Knowing that makes pushing anyone else (the "Fat man") - regardless of how s/he is described - off the bridge a despicable act of cowardice! It would be presumptuous and unconscionable. What a horrible predicament for "You"! If "you" go ahead and push the the "Fat man" to his death, you will live with that knowledge for the rest of your life, knowing you could have saved the children AND the "Fat man" by jumping yourself.

It seemed clear by the end of the show (especially when discussing whether to sacrifice one's own kin to save unknown kids) that our abstract moral rationalizing is rather a thin veneer over much deeper drives which actually determine people's choices. For example, that our moral sense is essentially motivated by empathy, and that we don't like taking obviously violent actions we would not want perpetrated on us (being pushed involuntarily into the path of the trolley). Using a mechanism (a switch and trapdoor) engages our rational faculties, while pushing the fat man directly elicits emotional revulsion. That we change our calculus for kin versus non-kin makes a pretty strong case for an evolutionary basis for these emotional drives on our behavior.

I am always (if not in all ways) entranced with human expressions of opinion and ideology. Sacrifice is usually never altruistic. This is true on both a conscious and genetic level: everything wants something for the least expenditure of effort (energy) and resources (available capital or stored fuel.) Easier is better. There have been numerous treatises written on this, yet most of us choose to pursue convoluted and difficult paths, because old paradigms say overcoming adversity builds character. That which does not kill you makes you stronger. Hogwash. That which does not kill you wastes time and energy you could have devoted to better endeavors. And that creates, among other things, the dark side of science---a different post, certainly---but related, absolutely.

There are few true dichotomies in life? and the decision to push a man onto the tracks in practice is a collective decision. The question of a morally justifiable sacrifice, as the trolley problem was originally framed, fails to acknowledging the underling plurality of solutions which arise with each unique situation.
Provided information about the trolley's potential destinations (probabilities), knowledge of the tracks involved, and experienced engineers, can we estimate the time required for a work around? A moral trolly conductor will (assuming he is committed to transporting individuals without injury, death, or a speeding ticket) slow the trolley down deliberately to give engineers the time they need to implement a solution.
After an estimate has been provided, and all attempts have been made to respond to the situation, will the contemplation of pushing men onto tracks become moral. It is immoral to inflict potentially unnessisary harm if YOU are not responding to the situation with the urgency you would expect if your own life was on the line. Free riding isn't reserved to passengers if the trolley is transporting for the benefit of your community.
The time before the fork (and collision) would be more appropriately visualized using an Olympic curling analogy (its a team sport).
If you need to push this fat man (please don't call him fat, he's sensitive)
onto the tracks for other reasons, then do it after the trolley has passed.

Trolley problems and other contrivances are unhelpful if we are truly interested in resolving ethical questions. If we look upon real-world occurrences and outcomes, we can learn much more, and, in time, reach useful conclusions about where we went wrong---unless we are encased in the intractabilities of historionicity that immobilized us in the first place. When I was a child, someone found a Roy Rogers cowboy glove I had lost on a playground. I had the other glove, found out who had its mate, proceeded to retrieve my property from his desk during recess, and was chastised for "theft" of my own property. That all got sorted out and ultimately, the true thief lost---I got my glove back. A small thing, sure---but important to a five-year old who had been taught justice.
Way now, sixty years later, this world is lost in its own intractability. Historionicity reigns continent to continent, and it is only partly due to religious differences. Culture, an outgrowth of religion, is far more dangerous. And greed? certainly. I wanted my glove back. Others want more---to which they have questionable rights.
You can't always get what you want. But---You won't always want what you get. Heh.

This post and the one concerning morality and revenge, fired several neurons. The five-year old (who was more probably six or seven) wanted justice. Most of us can relate to that sort of idealistic expectation. But, as we are all aware, people more frequently than not don't act right. I do not believe there is morality in acts of revenge. A current television series capitalizes on greedy games for greedy people, including the vapid little tramp known as Emily. Well, it is "entertainment", after all, and if it gets your adrenalin flowing and your heart rate up, your doctor should be pleased! Maybe---but you still need to go to the gym.
Sorry. I digressed. Sort of. I agree with a previous assessment regarding trolley problems and such like. What ifs do not translate to what ares. I have never intentionally killed anyone. Hopefully, I'll never be forced to make that decision. But, we just don't know, and, justice is a powerful motivator, yes?