As is so frequently the case, it's all a matter of definitions. While "a cyborg has unremovable implants in the body" certainly has a long pedigree, it's arguably less interesting in practice than "a cyborg is a person whose brain has adapted to using artificial intelligent tools as an extension of itself". (Note I don't mean "tools with AI" in the usual sense, just tools that themselves bring some intelligence to the party, however limited it may yet be.)

Neither is "right" or "wrong"; they're both just definitions.