The correct form of Maite
as nickname of María Teresa
, not Mayte
- is a usual form of María when it appears in Spanish nicknames, as in Maica, María del Carmen
, e.g. And the compound name is María Teresa
, not María y Teresa
; moreover, in compound names the prepositions and/or articles don't form the nickname: María de los Dolores
The spelling with Y is not correct in Spanish, but it is possible to find it just as other misspellings with Y: Aracely, Lydia...
In Basque, the name Maite means "beloved", not "to love" (you can check at Euskaltzaindia site: http://www.euskaltzaindia.com).