回答
This is not a piece of advice to you I'm talking to myself for my practice of writing skills. I've searched the phrases "a member of" and "a part of" on weblio and Collins dictionary. these phrases are not in the CD. only on WD I'd find the phrase "a member of" and "a member of". but these meaning'd be not far from the main word 'member/part' to the extent we can't imagine. I can't find any nuance other than the meaning of member/part itself. this doesn't mean that I'd insist any nuances aren't there. anyone could be able to say I can feel. I suppose it'd be good also there are as many opinions as there are people.
weblio.
https://ejje.weblio.jp/content/member
https://ejje.weblio.jp/content/part
https://ejje.weblio.jp/content/a+part+of
https://ejje.weblio.jp/content/a+member+of
collins.
https://www.collinsdictionary.com/dictionary/english/member
https://www.collinsdictionary.com/dictionary/english/part
この質問に回答する
「私は美術部に所属しています」
1) I'm a member of the art club. 2）I'm part of the art club.
これら2文のニュアンス(意味合い)の違いを教えて頂けましたら幸いです。