In US English (I can't speak for non-US versions) the reflexive would be assumed with "move," unless there was additional information to say otherwise.
Some other verbs are similar - "They dress," "They shave," for example. Some other verbs can work that way, but aren't necessarily as unambiguous.
I think this is the transitive/intransitive verbs divide: some verbs must have an object, other verbs normally do not. In English, if you "fall", you normally do not fall anything. "Fall" is a complete action, as is "move". On the other hand, if you "take", you must specify what you take. The sentence "John takes" is imcomplete. In Dutch, if you "bewegen", you must bewegen something, in this sentence "zich"(themselves) because the sentence is otherwise incomplete. These are simply conventions of language and not necessarily very logical. In English, you can "read" without specifying what you are reading although logically, you must be reading something, but you cannot "have" without detailing what you have. The short of the long of this is that Dutch "bewegen" must have a subject, and if they move without moving anything else, language require that we specify that they move themselves.
No, English suffered a transformation during the black death era. It almost became a creole language. It lost many features of the germanic languages, including reflexives features. It happens when a lot of people die and the remaining living ones forget the language suddenly. Or you are too lazy to learn the language you are supposed to speak, whatever happens first.
This is an interesting article about some of the peculiarities of English and how they developed.
No problem when you don't get this yet. You will get it while moving ;-) on with more exercises. Let me show you some:
Some Dutch verbs can only be used as reflexives, but the majority of the verbs can only be used non-reflexive. A lot of the verbs can be used in both forms. The Dutch verb 'bewegen' can be used in the reflexive form: 'zich bewegen' as well as the non-reflexive form 'bewegen'.
- Lets suppose your feet have been very cold and are warming them up. You want to tell me about it in Dutch:
'I am moving my toes'.
Using the reflexive form of the Dutch verb bewegen: 'zich bewegen', would then not be correct and would result in complete nonsense: like: 'Ik beweeg me mijn tenen.' (I am moving myself my toes). The correct translation would be: 'Ik beweeg mijn tenen'. Simply that! No more. You are informing that you are moving your toes.
- Now the next context: lets say you had an accident and were not able to move your toes for a while. Today you notice that you have recovered and that capabilities are returning. You want to describe this sensation to me:
'I'm moving my toes (all by) myself!'
Here again the application of the reflexive form 'zich bewegen' would be incorrect. You have to apply the non-reflexive form: bewegen. You are still just moving your toes. And that indeed, you are doing all by yourself. As you were stressing in this sentence: 'Ik beweeg mijn tenen zelf!
- But unfortunately you were mistaking. The movement of your toes was merely the result of some spasms:
'My toes keep moving (by themselves)!'
Reflexive use of the verb (zich bewegen) now would be appropriate: 'Mijn tenen blijven zich bewegen!'
There's nothing to it really ;-)