When I was being taught Mandarin Chinese- quite a few years ago- we were only taught 家人 as meaning family or family member. However Google translate returns 家庭 as the best option for 'family' and 家庭成员 for 'family member'. What is the distinction between these? Are they used in different contexts?
'家人' is a specific term for 'one's family'
For example: '我的家人在香港' (my family is in Hong Kong)
家人 can also refer to: 'individual(s) of one's family'. For example: '他是陳先生的家人' (He is Mr.Chan's family)
'家庭成员' is a specific term for 'member(s) of a family'/ family member(s)
We do not say: " My family members are in Hong Kong." in English; therefore, we don't say: "我的家庭成员在香港." in Chinese neither.
Similarly, we do not say: "He is Mr. Chan's family member" in English; therefore, we don't say: "他是陳先生的家庭成员" in Chinese neither.
We use the term '家庭成员' the same as 'family member(s) ' in English.
"Family members of the Chan family includes: Mr. and Mrs. Chan and their two daughters" (陳家的家庭成員包括：陳先生及夫人及其兩個女兒)
If you say: "Mr. Chan's family, are his wife and two daughters" then the translation would be: "陳先生的家人是他的妻子和兩個女兒”"
If you say: " Mr. Chan's family include he himself, his wife and two daughters" then the translation would be: “陳先生的家庭包括他自己，他的妻子和兩個女兒”
家庭 is a specific term for "Family"
家人 is a commonly used to describe close and family members, that have direct bloodline connection. 家族 resemble a whole "tribes". i.e. 家人, 家里人，家族人员。
However，成員 actually origin from Japan. i.e. 成員.構成員 メンバー(Meh O Bah [sounds like English member isn't it]), 家庭成员／家族の構成員。 Not forgetting Japan かてい(Kah Ten) 家庭.
Basically, you wouldn't see 成員 before the 18th century, nor 家庭. Chinese in various dialect rarely use 家庭成员 to describe the family members. In fact, 家庭成员 mostly used inside modern Chinese government document(China/Taiwan/HK ).
So it is awkward to say 我的家庭成员, Even though Mandarin speaker understands what you mean, but we will not use it.
Be careful that "家人" always means close ties than the word "兄弟"(brother, brotherhood).