The term first-generation, as it pertains to a person's nationality or residency in a country, has two incompatible meanings:
A native-born citizen or resident of a country whose parents are foreign born, or a foreign-born citizen whose parents immigrated when that person was very young, that is, the first native-born generation.
A foreign born citizen or resident who has immigrated to a new country of residence, that is, the first generation to immigrate.
This ambiguity is captured and corroborated in The Oxford English Dictionary's definition of "generation":
...designating a member of the first (or second, etc.) generation of a family to do something or live somewhere; spec. designating a naturalized immigrant or a descendant of immigrant parents, esp. in the United States.... (OED definition of "generation," section 6b., the term "first generation" is used to refer to foreign-born residents (excluding those born abroad of American parents).
There is no universal consensus on which of these meanings is always intended.