You are correct about FAMINC being a specific response on the CPS questionnaire and HHINCOME being the sum of INCTOT for all household members. The note about HHINCOME being the sum of all “adult” household members comes from the universe for INCTOT, which (since 1980) includes only those age 15 and older. HHINCOME is defined as the sum of INCTOT for all individuals within the household. The means that individuals not in the household at the time of the survey will not be included in the sum of HHINCOME. FAMINC may include income from family members not in the household at the time of the survey, but this will depend on the respondent’s judgement. Finally, HHINCOME is a continuous measure of income and FAMINC provides income categories.
You are also correct that not all household members are necessarily family members. This is what the FAMINC variable tries to identify. It asks the householder (e.g., the person filling out the survey), to report the total income of each member of their family. For those individuals who are not a member of the householder’s family, their income is not counted in the calculation of FAMINC. With that said, FAMINC is a “household level” variable and so all members of the same household will have the same value for FAMINC. Individuals not a member of the householder’s family have the householder’s family income for their observation in the FAMINC variable.