My answer to 1. is that Atyiah-MacDonald probably gives the right definition. Matsumura, like most other books I know, only treats primary decomposition for Noetherian rings. In this case things are simpler: for example the set of primes appearing in a minimal primary decomposition is the set of associated primes, something that fails in the general case. See this question for more details.

Atyiah-MacDonald, instead, gives the uniqueness theorems without the Noetherian hypothesis (although they treat the case of modules only in the exercises). For this reason, some definitions are slightly different. When everything is Noetherian of course there is no difference, but without Noetherian assumptions, I would stick with the definitions of Atyiah-MacDonald.

My answer to 1. is that Atyiah-MacDonald probably gives the right definition. Matsumura, like most other books I know, only treats primary decomposition for Noetherian rings. In this case things are simpler: for example the set of primes appearing in a minimal primary decomposition is the set of associated primes, something that fails in the general case. See this question for more details.

Atyiah-MacDonald, instead, gives the uniqueness theorems without the Noetherian hypothesis (although they treat the case of modules only in the exercises). For this reason, some definitions are slightly different. When everything is Noetherian of course there is no difference, but without Noetherian assumptions, I would stick with the definitions of Atyiah-MacDonald.