- If dualists saw someone saying that she/he is pain, they would think that their mental feels the pain but the physical doesn't feel the pain. They think the person is in pain because the mental feels the pain not because its body feels the pain and vice versa. They believe even if the body shows signs of pain, it's not from its mental.
This is essentially correct. The (substance) dualist believes that the final act of perceiving the pain is performed by the mental substance. The pain travels from the tooth through the nerves to the brain. At some point in the brain, it reaches a terminus, where it is perceived by the purely mental aspect of the mind. Dualists argue that this has to be the case, because of qualia. Here's an example that demonstrates what qualia is: Imagine a dentist called Mary, who is the top dentist in her field, she has aced every dentistry topic there ever was, can cure any patient who comes to her with a dental problem no matter how bad that patient's condition, and has studied every biological, neurological and chemical aspect of what a toothache is. In short she knows everything there is to know about toothaches. However, she has been an avid tooth brusher ever since she was a kid, and she has never ever had a tooth ache in her life. A dualist will argue that because of this, she doesn't really know what a toothache is at all. She knows all there is to know about toothaches, but she doesn't know what a toothache is, having never had the feeling herself. This difference between knowing everything about something, and knowing what something is or how it feels, is what philosophers of mind call qualia. Dualists think that qualia is proof that there is a non-physical dimension to the mind, since if the mind were purely physical, there would be no difference between the "about" and the "is".
- A behaviorist can identify that the person is in pain by observing her/his having painful facial expressions and having one of her/his hands on chin. Also by hearing her/his down tone.
Again correct. The behaviorist is coming from a different stand point, that ideas about qualia and dual mind substances are "unscientific" in the sense that whether they exist or not cannot be observed or proven. A theory is only as good as the predictions it makes, and qualia/dualism don't make any predictions whatsoever that a psychologist couldn't make based on behavior alone. The motivation for the behaviorist approach is mainly motivated by a desire to turn philosophy of mind into an empirical science that makes predictions. Behaviorism failed because it fell victim to the very same problem it tried to solve: it itself turned out to be "unscientific", because it had no predictive power either. Any behavior can be explained by multiple psychological explanations: The behavior of a drug addict, or a religious fanatic, or a person with schizophrenia, or someone just trying to fool the world, would all be the same, and a behaviorist has no way of telling the difference. To solve this problem, behaviorism has since been updated to a position called functionalism, which is similar to behaviorism, but it also admits that internal mental states do have a meaning (hence allowing to tell the difference between the examples I gave above of addict, fanatic, trickster, etc...)
The Identity Theorist would believe that one of parts in their mental is connected to the body part which feels the same as the part of the mental feels. Therefore, I think this is the most compelling position since this is how most people believe how our bodies work/function.
This is incorrect. The identity theorist would believe that there isn't any mental at all. Once the toothache signal leaves the tooth and moves into the brain cells, that electro-chemical signal is the pain. The pain has no other definition other than being the neural signal generated by a toothache. There is no separate mental state that the body "connects" to in any way, the mental state is identical to the neural state, hence the name identity theory. If we go back to our example of Dr Mary, the identity theorist would say that by knowing everything about toothaches, she knows was a toothache is, there is no difference, and qualia don't exist. The original identity theory of the 1950s was considered too strong: Since the concept of toothache pain for them was identical to the concept "human neurons c19 firing in response to tooth infection" that made the concept of toothache too strict. A dog or a chimpanzee would experience toothache pain just as much as human, even though their neural structure was entirely different. The more contemporary versions of identity theory are also called reductive materialism. In this version of identity theory, the mental state "toothache pain" can be reduced to various physical (neural) states of dogs, humans, chimpanzees, etc, and so is ontologically identical to those states. The concept still remains useful as a way of classifying and describing these states.
Note that dualism and identity theory are the only positions contradicting each other. Behaviorism/functionalism is often presented as being in contradiction with the other two, but it actually isn't, a functionalist can be either a dualist or an identity theorist. See this discussion:
What is the difference between functionalism and property dualism?
And the link provided in one of the comments there:
Also, you might want to look into eliminativism, a position which is different from any of the 3 you mentioned.