Where there is no meaning for a word or concept, it may be because it is implied. Where there is a shrap contrast between two meanings, it may be because we are wearing blinkers in expressing the meaning.
For the first point, if someone were to say to you, “I am your colleague”, and “We work together”, are they making 2 points or 1. Of course it depends on the meaning of “work” (be employed or do a task). If somebody (a onebody) says, “I am your mother”, “We are in the same family”, is she saying 1 thing or 2? Does, “I am a manager” imply the existence of staff.
You are wondering how a small group of a household can be identified as being a biological subgroup. Father, mother, parents and τέκνα implies what English declares as family. Kinship terms imply kinship.
For the point about seeming contrasts, settling in an area means making and accumulating wealth there, as well as just physically residing there (a shell we sleep in and then commute to the office). Washing clothes in a good way is a means of wealth preservation as much as it is form of hygiene. The verb οἰκονομέω or not just arrange the running of the household for it to function smoothly, but also to increase the wealth and output of the household. With slaves integral to means of production in many cases, they were of course included in the “household” - dwelling together and functioning together economically. Dwelling in a settled way for a long time was a way to accumulate more precious forms of wealth, such as paintings, bronzeware, etc.
If you are trying to map a western neuclear “family” onto οἰκία, it might sort of work. As has been mentioned, it wouldn’t be understood in the terms that you might want it to be understood, though. If you explained that the family was too poor to have slaves, the house was small and couldn’t accommodate uncles, aunts or grandparents and the house had no arable land, so not only the father, but also the mother had to work out as day labourers, then that might be okay.
