The compact form here, bija plus name plus namaha, is comparatively late as a text: the Ganapati Atharvashirsha that frames and popularizes it is usually placed in the sixteenth or seventeenth century, far later than the god it addresses, whose worship and iconography are attested from the Gupta period a thousand years earlier. What the late text did was give a devotional and philosophical frame, identifying Ganesha with Brahman itself, to a figure who had already been invoked informally at the start of undertakings for centuries. Transmission since has been unusually open by the standards of Tantric material: unlike many bija mantras that traditionally required initiation before a teacher would explain them, gam ganapataye namaha circulates in printed prayer books, on classroom walls and at the opening of public functions with no expectation of prior initiation, which tracks Ganesha's status as the deity invoked regardless of sectarian affiliation.
The translator's dispute is about the bija itself, gam. One position, common among Tantric commentators, holds that seed syllables are not words at all and cannot be glossed, only correlated with a deity by convention, so that gam simply is the sound form of Ganesha's presence and any English rendering falsifies it by making it look like vocabulary. The opposing position, more common in popular and pedagogical explanation, derives gam informally from the first sound of Ganapati and treats it as an abbreviated name rather than a non lexical seed, which is easier to teach but is regarded by traditional Tantric grammarians as a folk simplification of how bijas actually function.