The question title says it all—shall we rename the tag "movement" to "movado" (which is Esperanto and more specific to the Esperanto movement and distinguishing it from other movements)?
Since it is referring to the Esperanto movement, I would use movado.
I am not totally sure we need such a tag, but between movement and movado it is preferable the later. The former is too generic, and it should be replaced by esperanto-movement; since esperanto should not appear on our tags, movado is the only alternative I see.
It seems strange to me that just this one tag should be translated to Esperanto and thus movado. I personally think we should only have English tags unless we want to translate every tag and have twice as many tags for every question. Unless I'm missing something, if the tag movement is used in the Esperanto StackExchange, it certainly refers to the Esperanto movement. I'm against using the tag movado.
The word movement is vague, it can be a physical movement, or any kind of movement. With the word Movado it is clear that it is about the social movement that aims to propagate Esperanto in the whole world. The "Esperanto-movement" in English can be not understood as "movado", it can be understood as "everything in Esperanto", which is not good as we already have more precise tags for that.
Since the tag is too vague, I suggest that we divide movement into at least three tags (examples) :
- movado, for questions about organisms and methods to propagate the use of EO
- culture, for questions about the Esperanto culture
- speakers, for questions about Esperanto speakers (demography, fields of usage etc.)
This way, we will benefit more from the precision given by the tags.