According to Christopher Tolkien, his father had corrected the genealogies of 1959 (I believe this happened in 1970 or afterwards) by entering Anaire "as the wife of Fingolfin, with the note that 'she remained in Aman'" (quoted by HoME vol 11, p 323).
Something interesting is that Tolkien had also used this name for the wife of Turgon, whom he had originally named Alaire, before he finally decided on Elenwe. (I guess the similarity of the names finally got to him!)
So, the only conclusion can be:
Anaire: the wife of Fingolfin who remained in Aman.
"Fingolfin's wife Anaire refused to leave Aman, largely because of her friendship with Earwen wife of Arafinwe (though she was a Noldo and not one of the Teleri). But all her children went with their father" HoMe vol 12, p 344-345.
Arafinwe was the Quenya name given to Finarfin by Finwe.