Some of your scenarios can be handled relatively easily:

A person doesn't move because they're in shackles.

A person doesn't move because they've been injected with a drug that paralyzes them, but they are fully conscious and in control of their thoughts.

In these cases a person is physically unable to move. This raises no particular problem for saying the person acts on his motivations. The means available for him to act on his motivations are restricted.

The next set are not that much more difficult to understand:

A person doesn't move because at a young age they have been taught that moving in that particular time and place is evil and they will go to hell for doing so.

A person doesn't move because someone has threatened to harm their loved ones if they move.

The person has clashing sets of motivations. None of the options presented to him convince him. So he has to pick some irrational means of resolving the conflict such as ignoring criticism. He values one of his motivations and is willing to sacrifice rationality to it. So there is an explanation of his choice in terms of values.

You mention children:

At what age are children considered to start acting according to their own motivations?

Children act on their own motivations from sometime shortly before or after birth. This happens well before the child starts speaking. Language is a complex skill that can be only be acquired by thought and creativity, which is motivated action. People like to pretend children have no motivations for a couple of reasons.

Motivations acted on by ignorant people raise difficult problems and children are extremely ignorant by adult standards. So a child may act in a way that doesn't produce the result he wants out of ignorance. Adults are far too impatient to deal with this rationally and prefer to coerce children. The excuse for doing this is that the child doesn't really act on his motivations. That this is a lie can easily be figured out since the adults act exactly as if they were thwarting a person with motivations they don't like. The adults use guilt, threats of violence, confinement and theft. Such threats wouldn't work unless the child had motivations.

Your next set of examples are similar:

A person doesn't move because they have been hypnotized.

Are people under the influence of drugs acting according to their own motivations?

What about people with psychiatric conditions?

A person will often claim he had no motivation for some action he wants to disown. This is a lie. He has a motivation but wishes it was different, or wishes to con others into thinking it is different. So he claims he was in a state where he could not refuse to take some action as a result of drugs or hypnosis or mental illness. The problem isn't that he lacks motivation but that he wants to obfuscate it and lie about it.

Person A will often claim person B has no motivation if A wants to coerce B through threats of violence, confinement, forced drugging: the sort of stuff psychiatrists do as a matter of course. Person A might sometimes also offer B money or food or accommodation if B acts the way A wants. A might sometimes wants B to be dependent because it feeds A's vanity to have people he can help or ruin on a whim. Bribes and threats work precisely because people do have motivations, so employing them gives the lie to psychiatrists' claims that mental patients have no motivation.

To understand the situation with mental illness properly see books such as "Insanity: the idea and its consequences" and "The meaning of mind" both by Thomas Szasz.

I haven't gone through every situation you listed. You should try to work out the rest yourself.

You write:

Since mind and body are one and the same, there is no difference between manipulating the body and manipulating the mind, and so physical coercion and psychological coercion are no different from each other. How then can we define free agency?

I'm not sure what physicalism means. Thoughts are abstractions. Abstractions can be instantiated in any physical object that can perform particular operations. The set of operations required to instantiate all possible abstractions that can be copied, which includes all thoughts, is extremely limited, e.g. - Frieman and Toffoli gates. So any physical system that instantiates that limited set of operations may act in ways that can be explained in terms of abstractions. See 'The Beginning of Infinity' by David Deutsch, the chapters on abstractions and universality.

You haven't said what counts as physical coercion. Force, theft, fraud deprive people of the ability to do stuff. Threats of such actions are threats to deprive people of the ability to do stuff. So there is a difference between force, theft and fraud and threats thereof and other actions that makes them more problematic and limit a person's choices unless he is willing to suffer those force etc.