"Home", as an adverb, is often rendered in French by a phrase made up of the preposition "chez" and of a pronoun (moi, toi, lui/elle/soi, nous, vous, eux). The particular pronoun used to refer to somebody's home is chosen according to the following scheme.
- home of the locutor: moi. ex.: I'm going back home. → Je rentre chez moi.
- home of the interlocutor, second person sing.: toi. ex.: You are going back home. → Tu rentres chez toi.
- home of the person one is talking about, third person sing. : lui/elle. ex.: He/she is going back home. → Il/elle rentre chez lui/elle.
- home of a person one is talking about as being anyone, third person sing. : soi. ex.: You have to go back home at the end of the day. → Il faut rentrer chez soi à la fin de la journée.
- home of the locutor and his/her family, : nous. ex.: We are going back home. → Nous rentrons chez nous.
- home of the interlocutors : vous. ex.: You are going back home. → Vous rentrez chez vous.
- home of the persons one is talking about, third person plural : eux. ex.: They are going back home. → Ils rentrent chez eux.
There is not in French the notion of family attached in any special degree to "chez moi", "chez toi", etc., although this notion is not dissociated from it; in fact you can use these phrases in a quasi synonymous way to say "in my family", "in your family", etc.
- Chez moi/nous ça ne se passerait pas aussi bien, mon père se mettrait tout de suite en colère. (In my family that wouldn't do, my father would become angry right away.)
The sole notion is that of steady occupation of the place, coupled with a sense of belonging, and the notion of a certain degree of freedom enjoyed in that place by the person occupying it; they don't have to own the place, they can be renting it, and it can even be a hotel room, but people are not likely to say "chez moi" if, for instance, they have accomodations for the summer in a private home or an inn. It follows that "chez moi", etc., is equivalent to "my place", "your place", etc. as used by people who are the tenants or owners of a house, flat/appartment or room.
(I think, however, that "home" is not totally dissociated from the idea of "place where you live" in those cases where you are on your own and not surrounded by a family.
To the term "place to live/place" corresponds "chez soi", which is also a noun; "chez moi", "chez vous", "chez nous", "chez eux" are also used.
"À la maison" is another phrase meaning "home" (adverb); it is synonymous with the expressions discussed above; you can see in the third example below that it means "home" in the sense of "place of living of the family".
(TLFi − Loc. adv. À la maison. Chez soi.
♦ Aller, être à la maison; (aimer) rester à la maison.
♦ J'ai éprouvé un sentiment de malaise, qui ne s'est calmé que quand je suis rentré à la maison, où je me suis promené en tout sens, pendant près d'une heure (Delacroix, Journal, 1853, p. 50).
♦ Son jeu principal consiste à marcher dans le ruisseau (...) ce qui lui attire des calottes à la maison (Verlaine, Œuvres compl., t. 4, Mém. veuf, 1886, p. 288).
"Foyer" or sometimes "foyer familial" is another term that can be used to render the noun "home" as the noun, but it is rather literary and not used much or used in certain phrases (femme au foyer).
The first sentence, given the context, could be rendered as follows.
((a) “My home isn’t a home, it’s just a house.”)
- Mon habitation n'est pas un logis familial, ce n'est qu'une maison sans maisonnée. (TLFi) (Note that, according to what has been said, "un chez moi" cannot be used to much effect here.)
((b) “My home isn’t really a home, it’s more like/of a house.”)
- Mon habitation n'est pas vraiment un logis familial/foyer (familial), c'est plutôt une maison sans vraie maisonnée.