As usual in topology, there are many different definition for terms like "locally connectedness":
Let $(X,\mathcal{T})$ be a topological space. Note that I use the following definition of neighbourhood: A set $U\subset X$ is a neighbourhood of $x\in X$, if there is an open set $\mathcal{O}\subset X$, such that $x\in\mathcal{O}\subset X$. Therefore, neighbourhoods can also be closed.
(1) $(X,\mathcal{T})$ is called "weakly locally connected" at $x\in X$, if for every neighbourhood $U\subset X$ of $x$ there is a connected neighbourhood $V$ of $x$, such that $x\in V\subset U$. In othere words $x$ admits a neighbourhood basis of connected sets. If $(X,\mathcal{T})$ is weakly locally connected at every $x\in X$, then it is called "weakly locally connected".
(2) $(X,\mathcal{T})$ is called "locally connected" at $x\in X$, if for every open neighbourhood $U\subset X$ of $x$ there is an open connected neighbourhood $V$ of $x$, such that $x\in V\subset U$. In othere words $x$ admits an open neighbourhood basis of connected sets. If $(X,\mathcal{T})$ is locally connected at every $x\in X$, then it is called "locally connected".
The two definition differ just by the word open.
Obviously, if $(X,\mathcal{T})$ is locally connected at $x\in X$, it is also weakly locally connected at $x$. The reverse is in general not true. However, we can show that every weakly locally connected space is also locally connected. Therefore, the two definition are globally equivalent. (For a proof see https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Locally_Connected_Space)
I am wondering, why there is no such thing as a weakly locally path-connected space. I have found both definition of locally path-connectedness in textbooks: every neighbourhood contains a path-connected neighbourhood and the second one with open neighbourhoods....are they in this case already equivalent locally? Also Wikipedia doesn't make a distinction: They define weakly locally connected and locally connected, but they define only locally path-connected without a weakly version... (https://en.wikipedia.org/wiki/Locally_connected_space)
In other words: Is the following statement true:
Let $x\in X$ be fixed.
Every neighbourhood $U$ of x has a path-connected neighbourhood $V$, such that $x\in V\subset U$ $$\Rightarrow$$ Every open neighbourhood $U$ of x has an open path-connected neighbourhood $V$, such that $x\in V\subset U$
That the statement is again globally (if we assume that both sides holds for all x) true is obvious, by a similar proof as for locally conectedness..... But I guess that is also true at a certain point, because otherwise there would also ba such a thing as weakly locally path-connectedness....