What is the definition of "sheet"? Here is a definition for evenly covered set.

Definition
Let $C,X$ be topological spaces and $p:C\rightarrow X$ is a continuous function.
Let $U$ be an open subset of $X$.
Then $U$ is said to be evenly covered by $p$ if there exists a family $\{V_i\}_{i\in I}$ of mutually disjoint open sets in $C$ such that $p^{-1}=\bigcup_{i\in I} V_i$ and $[\forall i\in I, p\upharpoonright V_i:V_i\rightarrow U$ is a homeomorphism.

I want to know what of below statements is the definition of "sheet".

Let $C,X$ be topological spaces and $p:C\rightarrow X$ be a continuous function.
Let $U$ be open in $X$ that is evenly covered by $p$.
Let $V$ be open in $C$.
Then (1)$V$ is said to be a sheet over $U$ if $V\cong U$.
(2) $V$ is said to be a sheet over $U$ if $p\upharpoonright V:V\rightarrow U$ is a homeomorphism.

What between (1)&(2) is the definition of "sheet"?
Also, should $V$ be required to be an open set?
Plus, if (2) is the correct definition, then is it okay to call $V$ as "a sheet over $U$ by $p$" to emphasize $p$?

In the first definition you have for what it means for $U$ to be evenly covered, the mutually disjoint open sets are your sheets.
$(1)$ and $(2)$ say the same thing, except for $(1)$, you would probably add "a sheet over $U$  via $p$" (or "by $p$", as you said) to emphasize that the homeomorphism is given by $p$ restricted to $V$.
