I can explain it like this.
Given a set $X$, write $X^*$ for the collection of all finite sequences in $X$, including the empty sequence. Now let $L$ denote an arbitrary set, which we think of as a "language"; so you should be thinking of the elements of $L$ as "formulae". Then:
Definition. An inference relation over $L$ is subset $\vdash$ of $L^* \times L$ subject to certain axioms, like:
For all $\Gamma \in L^*$ and all $\varphi \in L$, if $\varphi$ occurs somewhere in $\Gamma$, then $\Gamma \vdash \varphi$.
Note that we write $\Gamma \vdash \varphi$ as a more readable alternative to the more correct $(\Gamma,\varphi) \in \;\vdash$. This notation can be seen in axiom 1, for example.
Now here's where the ambiguity creeps in. Suppose $L$ is the set of all strings featuring the symbols $0,1$, the comma symbol, and the symbol $\vdash$. So a generic element of $L$ looks like:
You can see the issue, right? If we suppose furthermore that $\vdash$ is an inference relation on $L$, then we cannot tell what "$01,1 \vdash 1$" means. It could be an element of $L$. Or, it could be the writer attempting to claim that $(\langle 01,1\rangle,1) \in \;\vdash$. Without further information, we cannot know.
To avoid this kind of ambiguity, we would choose a different symbol for the inference relation, as in:
Suppose furthermore that $\vdash'$ is an inference relation on $L$.
It now becomes clear that $01,1 \vdash 1$ is intended to denote an element of $L$, whereas $01,1 \vdash' 1$ is expressing the proposition that $(\langle 01,1\rangle,1) \in \;\vdash'$.