Do not try to find difference between Core HR and HR Generalist. Core HR are Functions or Responsibilities whereas HR Generalist is a Person.
Broadly following functions can be considered as Core HR Functions :
Manpower Planning, Recruitment, Employee Engagement, Training & Development, Employee Welfare, Personnel Administration, Performance Management, Compensation / Salary Management, Industrial Relations and Separation. ( I hope I have not missed out any).
Above Core HR Functions are performed in all organizations in varying degrees. Depending upon the size and kind of organization and depending upon the extent of organizational evolvement, there could be one or many Managers /Officers looking after above functions. Roles of such Managers /Officers could be specific such as recruitment or training & Development. These Managers / Officers can not be called as HR Generalists since they are handling only specific area/functions.
Whereas Manager or Officer handling most or all of the Core HR functions is HR Generalist.
To conclude HR Generalist is a person who is responsible for most of the Core HR Functions, in a organization.