Foliations with measure preserving holonomy, Author J. F. Plante, Annals of Mathematics, $102$, $(1975)$, $327-361$.
Page $337$ it is mentioned "If $x \in M$ then $x$ is contained is some plaque $\rho$. The leaf of $\mathcal{F}$ through $x$ is just the union of all plaques which occur in plaque chains starting with $\rho$".
The question is: we only take into account the plaque chains that start with the plaque $\rho$ or all the plaque chains that the plaque $\rho$ belongs to. In other words, we should even consider the plaque chains in which the plaque $\rho$ is just one of their elements (not necessarily the first plaque in the chain).
The plaque chains which catch $x\in p$ somewhere in the middle of the list are covered by two plaque chains that start with $p$. So taking the union over $(p_1,...,p_i,...,p_r)$ with $x\in p_i$ is equivalent to taking the union over $(p_i,p_{i-1},..., p_1)$ and $(p_i,p_{i+1},...,p_r)$. What Plante is trying to convey here is, I believe, that with the notation $\mathcal{F}(x)$ he is considering the leaf of the foliation $\mathcal{F}$ passing through $x$ and he is considering said leaf to be marked at $x$. $\mathcal{F}(x)$ is centered at $x$, so to speak; even though there is no natural center to any leaf, in general.
In general the formalism of foliations is often not optimized; I would suggest drawing pictures: for a foliated space or manifold, you have little boxes (= foliation boxes) with marked horizontal pieces (= plaques) (rectangles with horizontal lines for simplicity). By extending horizontal pieces from one little box to another one one gets plaque chains and the whole extension gives you leaves.