In the Vishnu Purana (Viṣṇu Pūraṇa, VP) we find the term self-born. Other “modes of birth” mentioned in the VP are mind-born, will-born and self-born.

Mind-born is used only in connection with the sons of Brahma, the seven rishis, or prajapatis. In VP 1.7.4 they are called literally “mind sons” or mānasāḥ putrāḥ. This term is translated (correctly) by H.H. Wilson as mind-born sons. HPB uses the term Manasaputras, and explains it as those who are “born of ‘Mahat’, or Brahmâ” (SD II, 167), or Mind-born sons (SD II, 374).

yadāsya tāḥ prajāḥ sarvā na vyavardhanta dhīmataḥ |
athānyānmānasānputrānsadṛśānātmano ‘sṛjat || VP 1.7.4||

Will-born is used in the same sense as mind-born, in connection with the rishis.

Self-born is used for example in connection with Vishnu and his hypostases. The element of apparition is not the essential characteristic here, but the idea of parentlessness, the idea that something appears from nothing other than itself, that is self-born.

HPB also uses self-born in the sense of parentless, for example in SD I, 109: “They [the Manushi-Buddhas] are the ‘Buddhas of Contemplation,’ and are all Anupadaka (parentless), i.e., self-born of divine essence.” Anupadaka would be erroneous for upapāduka, which is an adjective meaning self-produced, often used in the context of the modes of birth.