A standard terminology for different definitions of complete sublattice

Question

Let $(X,\le)$ be a complete lattice and $A\subseteq X$.

I'm trying to find a standard terminology for special types of sublattice. What is $A$ called if

1. $(A,\le_A)$ is a complete lattice.
2. $(A,\le_A)$ is a complete lattice such that all finite infimums and supremums in $(A,\le_A)$ are those in $(X,\le)$.
3. $(A,\le_A)$ is a complete lattice such that all infimums and supremums in $(A,\le_A)$ are those in $(X,\le)$.
4. $(A,\le_A)$ is a complete lattice such that all infimums and supremums in $(A,\le_A)$ are those in $(X,\le)$ and in addition $\min A= \min X$ and $\max A=\max X$.

It seems a complete sublattice can be any of (2), (3) or (4). Is there a standard terminology for definitions above?