I think that “chassidish” is a culture, a philosophical school and a movement and the three different kinds of chassidim are distinct groups that don’t have as much overlap as people think which is what lead to the confusion in this thread.
Culture- the “international” chassidim you see in BP and sometimes elsewhere. They enjoy the heimishe atmosphere, they like the fact that everyone around them is fully observant and medakdek bemitzvos but they don’t necessarily learn chassidish seforim and are more likely to have a rov (even if they call him a “roof”) than a rebbe.
school of thought- someone who learns chassidish seforim and tries to follow some of their teachings in his avodas Hashem. You don’t have to be culturally chassidish to do this. I learn chassidish seforim. I’m about as far from chassidish as you are likely to meet.
movement – He follows his Rebbe. Period. Zehu.