Alex, I can understand your perspective and there may be a considerable number who consider themselves “their whole body”, but I think most people have the conception that they are the consciousness that inhabits and wills the body.I believe that both the assumption that most people consider themselves to be something immaterial living in the body and, to the degree that some people actually do believe that, the reasons for the belief are wrong.
The illusion of separateness from our body I think is rather strong. You may argue that this is just an illusion and you would be completely correct. But I would respond with the concept of soul which seems to be a popular concept throughout the world, not universal but something quite like it probably is for a majority of the human race. This concept suggests that there is a separate conscious entity that is distinct from the body. Why is this an almost universal concept, well the illusion the illusion that we are a separate homunculus is very strong. So I think those who support the idea of the self being defined as the part of the body/mind that is conscious and self aware have the stronger case.
Another thought experiment that will put this in better perspective. It we were able to perfectly record and make an exact copy of your nervous system and then simulate it in a sufficiently advanced computer-like device, I would argue that the simulation would have an experience and that experience could be labeled “you”. Conversely, If we used an extremely potent general anesthesia that precluded any awareness yet did no damage either to your brain or body would “you” still exist?
Now obviously there are some people who have had out of body experiences, perhaps due to a major accident or under recreational drug use. I have known at least one such person myself back in Germany. But if we look around us I think we will quickly find that those who have such an experience and then conclude that they are an immaterial spirit or suchlike because they fail to ask the most urgent follow-up questions** are still a small minority of all people.
In fact, as exhibit #1 I wish to present myself. I do not identify with something immaterial inside of my body, I identify with my body.
As exhibits #2 to several billion I present most other people on this planet.
Make a thought experiment. Imagine you shoot somebody in the foot. What are they most likely going to say? Is it going to be
Ah, look at that, you shot my body in the foot. Good that that isn't really me because I am only using it as a vehicle. But I still have to ask, why did you do that?Or is it going to be
Argh! You bastard! Why did you shoot me in the foot???Well, I think I can rest my case at this point.
To the degree that people want to believe that we are really souls or whatever the motivation will rarely be that they personally do not identify with their bodies. Even the ones who very explicitly hold that belief would say "I have lost a leg" if it had to be amputated instead of "my body has lost a leg".
Instead what motivates them to at least believe to believe in souls is quite simply fear of death and, perhaps in particular, fear of the annihilation of their loved ones at death. That is very understandable but a very different cup of tea than personal identification as something apart from the body.
Here again, by the way, are interesting parallels between religious and non-religious dualists. The former are well understood: They hope that a soul lives on after death so that they and their loved ones are immortal in the afterlife. Non-religious dualists are often singularitarians or transhumanists who hope that their 'consciousness' could be transferred into a computer through mind uploading, as if it were a thing that you can move around. The hope in this case is that somebody will invent a sufficiently advanced computer before they die so that they and others can become immortal in cyberspace.
*) Really I see no relevant difference between a religious person who believes that we are supernatural immaterial souls and a non-religious person who believes that we are our consciousness, whatever that even means. Both are instances of dualism, only the second person is in denial about it. One of the reasons I do not see a difference is that I have yet to read or hear a meaningful definition of the term supernatural, and if you remove that term there is no real difference left anyway.
**) Top of the list should be "if I am outside of my body, what am I seeing things with?" and "if I am outside of my brain, how can I still think and remember?". Bonus points for "if I can think and remember while I am immaterial, how come that purely physical and chemical trauma or diseases can destroy these processes in other people?"