The paper reviews the definition of ‘internationalisation’ of higher education shaped by Knight and colleagues in the 1990s and shepherded through successive revisions, justifications and debates. Internationalisation is defined as ‘the process of integrating an international, intercultural or global dimension’ into post-secondary education. This conceptualisation underpins scholarship on international education and is used to draw together practitioners of cross-border education, primarily in the Anglophone zone and Western Europe. However, while the definition has received very widespread recognition and support, cross-border higher education has not evolved as its practitioners want and there is growing criticism of and disillusionment with the definition and its double claim to explain and unify. Its problems are fundamental. The purpose of shaping practice crowds out the scholarly mission to understand and explain, while its universalism conceals ambiguities that weaken the purchase on practice. It rests on an ideological binary between ‘globalisation’ (bad) and ‘internationalisation’ (good) that downplays global activity in higher education and science, and locks practitioners into the national container and hence into government policies in favour of competition and commercialisation. It is fundamentally non-relational, focused on the qualities of the self without regard for the consequences of self-internationalisation for the other, and this enables continued Euro-American (Western) centrism in cross-border higher education, in lineage with colonialism. The definition cannot adequately inform either scholarship or practice. The paper suggests an alternative approach to terminology and relationality in cross-border higher education.