The bonding in the layers of graphene are sp2 hybridised and it forms a giant covalent lattice. The bonds between the layers are considered metallic in character nd are weaker. IUPAC for example indicates their strength comparable to intermolecular forces. Nevertheless many authorities describe the structure of graphite as a giant covalent lattice. Understanding what’s going on is probably more important than what you choose to call it but in the UK the main exam boards, including Oxford, Cambridge and RSA expect to see graphite described as a giant covalent lattice even though that does not necessarily tell the whole story. So for students taking their exams it’s as well to be aware of their expectations. I would say you are not wrong in some of your thinking but be aware of the way various interpretations of the structure are described.