I saw the following definition for initial segment in https://digitalcommons.kennesaw.edu/cgi/viewcontent.cgi?article=2161&context=facpubs
If $\leq$ is a partial order in a set $X$, then a chain $C\subseteq X$ is a subset C of X that is totally ordered by the order $S$ . Note that the empty set is a chain. If $C$ is a chain in $X$ and $x\in C$, then we define $P(C, x) = \{ y\in C| y < x\}$. A subset of a chain C that has the form $P(C, x)$ is called an initial segment $C$.
But in wikiproof (https://proofwiki.org/wiki/Definition:Initial_Segment#:~:text=The%20concept%20of%20an%20initial,this%20concept%20as%20a%20section.), initial segment is defined as
Let $(S,\leq)$ be a well-ordered set. Let $a\in S$. The initial segment (of S) determined by $a$ is defined as: $S_{a}:=\{b\in S:b\leq a∧b≠a\}$
Are these two definitions equivalent? One defines $\leq$ to be a partial order, and the other defines it to be a well-ordering.