Two equivalent definitions of endpoints of continuum

Question

In different materials there are used different definitions of an $\it{endpoint}$ in a continuum (compact and connected metric space). Some of them are equivalent, but some of them have minor differences (see https://www.jstor.org/stable/1989267?seq=1).

I am interested in proving that the following two definitions of endpoints are equivalent:

1. Let $X$ be a continuum. We say that $p\in X$ is an endpoint in $X$ if for any two subcontinua of $X$ containing $p$ one of them must be contained on the other.
2. Let $X$ be a continuum. We say that $p\in X$ is an endpoint in $X$ if $p$ has arbitrarily small neighborhoods each of whose boundaries contains a single point.

Answer 1

Consider that old favourite the closed topologist's sine curve $S =\{<x,y> \in [0,1] \times[-1,1]:y = \sin(1/x) \text{ or }x=0 \}$ and the point $p = <0,-1>$. Any connected closed subset containing $p$ is either $\{0\} \times [-1,y]$ or $S \cap ([0,x] \times [-1,1])$ for some $x$ or $y$ so $p$ satisfies condition 1. It clearly doesn't satisfy condition 2, so they are not equivalent.