I have always felt that a Jesus-follower's second responsibility is to their family. Sometimes "family" is referred to as "loved ones."
My first responsibility is to love God.
My second responsibility is to love those in my family.
My third responsibility is to share the love of Jesus with others.
If I don't live out our first two responsibilities, my third responsibility will be inauthentic.
If someone claims to follow Jesus, but does not fulfill their second responsibility, why would I listen to what they say to me about the love of Jesus?
Dallas Willard puts it like this.
"A common usage of the word neighbor today locates the neighbor as one who lives “next door” or close by. [See Luke 10:36-37] A “next-door” neighbor is one with a special degree of intimacy, in this understanding, and there is something to that. But in this understanding my most important neighbor is overlooked: the one who lives with me— my family, or others taken in by us. They are the ones I am most intimately engaged with in my life. They are the ones who first and foremost I am to love as I love myself. If only this were done, nearly every problem in families would be resolved, and the love would spread to others."