different definitions of Hopf algebras

(i). In the book Algebraic Topology, A. Hatcher, p. 283， the notion Hopf algebra is defined as follows:

(ii). However, in the book Bialgebras and Hopf algebras, J.P. May, the notion Hopf algebra is defined as follows:

Question: why the definition in (ii) is much more complicated than the definition in (i)? Are the two definitions of Hopf algebra in (i) and (ii) equivalent or different? I do not understand the definition in (ii).

Another question: for an $H$-space (we can strengthen to topological monoid up to homotopy) $X$ and coefficient ring $R$, will the homology $$ H_*(X;R) $$ be a Hopf algebra according to the definition in (ii)?

[Note: After the comments to this answer and the question, there have been major changes. Thanks to Najib Idrissi, the situation is much clearer now.]

There are only slight differences in the definitions and strictly speaking, Hatcher's is more general. In addition to (equivalents of) Hatcher's axioms, May wants a Hopf algebra to be flat (a minor technical additional assumption) and the comultiplication to be coassociative. Furthermore, it remains unclear (to me) whether coassociativity is automatic in Hatcher's definition.

To make it clear, a Hopf algebra in the sense of Hatcher's is one in May's definition, if in addition

1. $A$ is flat as $R$-module and
2. $\Delta$ is coassociative, i.e., makes the left hand side diagram in May's definition 2.2 commute.

(Perhaps coassociativity is automatic, but I don't see why.) For how to define the antipode, see this blog post. The unit(=coaugmentation) should be taken to be the algebra structure map $R\to A$ and the augmentation $A\to R$ should be taken the projection $A\to A_0\cong R$.

The other way around, given a Hopf algebra $(A,\varphi,\psi,\eta,\varepsilon,\chi)$ as May wants it, then $\psi(\alpha) = 1\otimes\alpha+\alpha\otimes 1+\sum_i\alpha'_i\otimes\alpha''_i$ follows from the commutativity of the right hand side diagram in 2.2 and the fact that $A_0 = R$ via the unit and counit. The commutativity of the diagram in 3.2 as well as the second one in definition 2.2 show that $\psi$ is a homomorphism of algebras.

An example of a Hopf algebra as in Hatcher's which isn't a Hopf algebra as in May's definition would need to be non-coassociative (besides maybe non-flatness, which wouldn't be satisfactory). As I said before, I don't know if this is automatic or not.