The definition of two-step iteration in Kunen differs slightly from the definition in Cummings' handbook chapter section 7. Following Kunen, by a $\mathbb{P}$-name for a forcing poset I mean a triple of $\mathbb{P}$-names $\dot{\mathbb{Q}},\dot{\leq}_\mathbb{Q},\dot{1}_\mathbb{Q}$, such that $1_\mathbb{P}\Vdash$ $(\dot{\mathbb{Q}},\dot{\leq}_\mathbb{Q})$ is a pre-order with maximal element $\dot{1}_\mathbb{Q}$.
Kunen: The conditions in the iteration $\mathbb{P}*\dot{\mathbb{Q}}$ are pairs $(p,\dot{q})\in\mathbb{P}\times\text{dom}(\dot{\mathbb{Q}})$ such that $p\Vdash\dot{q}\in\dot{\mathbb{Q}}$.
Cummings: A set $X$ of $\mathbb{P}$-names is called a core of $\dot{\mathbb{Q}}$ if $1_\mathbb{P}\Vdash\dot{q}\in\dot{\mathbb{Q}}$ for any $\dot{q}\in X$, and whenever $\dot{y}$ is some name such that $1_\mathbb{P}\Vdash\dot{y}\in\dot{\mathbb{Q}}$, there exists $\dot{q}\in X$ such that $1_\mathbb{P}\Vdash\dot{y}=\dot{q}$. It can be proven that a core $X$ exists. The conditions in the iteration $\mathbb{P}*\dot{\mathbb{Q}}$ are pairs $(p,\dot{q})\in\mathbb{P}\times X$.
First, what is the purpose of $p\Vdash\dot{q}\in\dot{\mathbb{Q}}$ in Kunen's definition?
Second, Cummings says although the difference is essentially trivial, under Kunen's definition it may happen that $\mathbb{P}$ is countably closed and $1_\mathbb{P}\Vdash\dot{\mathbb{Q}}$ is countably closed, while $\mathbb{P}*\dot{\mathbb{Q}}$ is not. I understand that in general $\text{dom}(\dot{\mathbb{Q}})$ is not a core of $\dot{\mathbb{Q}}$, but haven't found an explicit example of Cummings' claim. Actually I wonder why the following proof doesn't work to prove that $\mathbb{P}*\dot{\mathbb{Q}}$ is countably closed under Kunen's definition: suppose $(p_n,\dot{q}_n)_n$ is a descending sequence of conditions. Let $p$ be a lower bound of $p_n$s. Then
$p\Vdash(\dot{q}_n)_n$ is a descending sequence of conditions.
Therefore,
$p\Vdash(\dot{q}_n)_n$ has a lower bound.
Strengthening $p$ or using maximal principle, we may assume there is a name $\dot{y}$ such that
$p\Vdash(\dot{q}_n)_n$ has a lower bound $\dot{y}$.
In particular $p\Vdash\dot{y}\in\dot{\mathbb{Q}}$. By definition of forcing relation, there exists $p'\leq p$ and $(\dot{q},r)\in\dot{\mathbb{Q}}$ such that $p'\leq r$ and $p'\Vdash\dot{y}=\dot{q}$. Therefore $(p,\dot{q})$ is a lower bound of $(p_n,\dot{q}_n)_n$ in $\mathbb{P}*\dot{\mathbb{Q}}$.