For a $C^*$-algebra $A$, G.J. Murphy (in his book page 38) Defines the multiplier algebra $M(A)$ of $A$ to be the set of its double centralizers.
In this book, they define $M(A)$ as the collection of all mapping $T:A \rightarrow A$ such that for any $x ,y \in A$, $x(Ty)=(Tx)y$.(Definition 1.4.10)
Another definition is This by the universal property.
Are there other definitions that are equivalent to the above or somehow related?
There are many equivalent definitions of the multiplier algebra $M(A)$. I'll list some, but I'll only describe the underlying sets.
$(1)$ The set of double centralizers $(L,R)$ which consist of maps $A\to A$ satisfying $R(a)b = aL(b)$ for all $a,b \in A$.
$(2)$ (Essentially the same as $(1)$) The set of maps $L: A \to A$ for which there exists a map $R: A \to A$ such that $R(a)b = aL(b)$ for all $a,b \in A$.
$(3)$ View the $C^*$-algebra $A$ as a right Hilbert module over itself in the obvious way. Then we can define $M(A) = \mathcal{L}_A(A)$, the $A$-adointable operators.
$(4)$ The multiplier $C^*$-algebra as the solution to a universal problem: $M(A)$ is the largest $C^*$-algebra that contains the $C^*$-algebra $A$ as an essential two-sided ideal, in the sense that if $j: A \to C$ is a $*$-morphism with $j(A)$ an essential ideal in $C$, there exists a unique $*$-morphism $M(A) \to C$ extending the map $j:A \to C$.
$(5)$ If $A \subseteq B(H)$ is a faithful non-degenerate $*$-representation (we can also represent on a Hilbert module instead of a Hilbert space), we can define $$M(A) = \{x \in B(H): x A \subseteq A \text{ and } Ax \subseteq A\}\subseteq B(H).$$
All these interpretations are canonically $*$-isomorphic. The way to see this is to show that each one of them satisfies the universal property satisfied in $(4)$ (though it is immediately clear that the interpretations $(1), (2), (3)$ give the same $C^*$-algebra). A good reference to see the equivalence $(3) \iff (4) \iff (5)$ (which is basically the interesting part) is Lance's book "Hilbert $C^*$-modules" chapter 2.