While I was reading definition of a transitive relation on the set $X$ on Wikipedia: $$\forall a,b,c:(aRb\land bRc)\implies aRc$$ I noticed that it uses logical implication instead of material implication, but isn't the logical implication a sort of "meta-concept"? If it is, doesn't it mean that we can't use it inside of definition in our theory (which can be taken to be set theory)? Or is it just that sometimes $\implies$ is used as material implication?
In the Wikipedia definition, you should take both the $\wedge$ and the $\Rightarrow$ as being formal logical connectives in the syntax of first order logic. They are not things at the meta level.
You are correct that sometimes $\to$ and $\Rightarrow$ are used denote quite different things, and that sometimes the distinction is between object level and meta level. There is no consistent convention on this and it is all very confusing for people learning about logic.
Short answer: Sometimes ⟹ is used as material implication.
Long answer: See e.g. my answer here: Implies ($\Rightarrow$) vs. Entails ($\models$) vs. Provable ($\vdash$) As I remark there, "⟹" is used in at a number of different ways, and is probably best avoided!