In this work, we relate the three main formalisms for the notion of pasting diagram in strict ω-categories: Street's parity complexes, Johnson's pasting schemes and Steiner's augmented directed complexes. We first show that parity complexes and pasting schemes do not induce free ω-categories in general, contrarily to the claims made in their respective papers, by providing a counterexample. Then, we introduce a new formalism that is a strict generalization of augmented directed complexes, and corrected versions of parity complexes and pasting schemes, which moreover satisfies the aforementioned freeness property. Finally, we show that there are no other embeddings between these four formalisms.