['Sympathizer'] I thought it is more inclusive than supporters, including those whose support is milder, more abstract and less decisive, at the level of emotional sympathy and nothing that goes beyond it.
That is correct - a good definition, although 'Sympathizers' can also express parallel emotions in various minor activities.
However, I googled it and had the impression that 'sympathizer' is mainly used derogatively (e.g. Nazi sympathizer)
"Sympathizer" is, itself, neutral.
(i) one who demonstrates or shares his emotional feelings about someone or an organization
(ii) one who shares the opinions/ideals of someone or an organization.
(i) is used in a more literal, emotional, and a positive sense in such sentences as
"The little girl died in hospital. Sympathizers left flowers and toys at the scene of the accident." and has the meaning of "people who wished to share their similar emotions with the victim's family,"
In (i), there is some action but it is indirect and personally motivated.
(ii) "People who think in some defined or implied way and who broadly and naturally agree with the aims of something/someone but fall short of being members or dedicated followers."
In (ii), 'sympathizer' is often used to imply a natural, mass agreement: the psychological effect on the reader of this catch-all term is "That is not me, but the threat is great!" and thus there is a "group whose thoughts are wrong" and this causes agreement with the writer's opinion.
Thus, with an organization that the writer thinks is abhorrent, 'sympathizer' is negative.
(Compare with the phrase "The Silent Majority" (https://en.wikipedia.org/wiki/Silent_majority, which was used to indicate a mysterious and undefined set of people who share the speaker's opinions and whose opinions are normal.)
My question is: can you also talk about 'sympathizers' of a movement without insinuating that it is a bad thing to support it?
"The call by the Mayor of Manchester for an increase in the state pension for the elderly has gathered many sympathizers, some of whom protested in front of the parliament building." These are “actual and emotional” supporters
“Members of the “Build Flood Defences Now” campaign have gathered many sympathizers who understand that many people in the area will lose their houses in the coming years, but who also see that the huge expense of building such defences is not really justifiable.” These are more “emotional, rather than actual” supporters.
The two nuances have overlap and the circumstances of a positive meaning with an organization are thus possible but rare, as positive sympathizers are those who understand the emotional feelings engendered by the problem but may disagree with the organization’s methods, solution, or overall aim.