people often define the composition of two functions $f, g$ as follows: $$ f\circ g(x) = f(g(x)). $$
then, if we already know an $x$ and we want to know the value of $f\circ g$, first we calculate $y=g(x)$ and then put this $y$ into $f$.
so if we have a multiple composition $f\circ g \circ h \circ \cdots \circ j$, we have to calculate from the right to the left.
in this notation, the chain of calculation operated from the right to the left.
but a few authors define the composition as follows: $$ \left<x,y\right>\in \mathrm{graph}(f\circ g)\iff\exists z, (xfz \wedge zgy.) $$
obviously in this definition every calculation starts from the left and goes to the right.
I would like to know which definition is better for the short and intuitive argumentation.
of course, so many definitions are just customs so the only thing we have to do is just to remember and to use that, at least for communication.
but sometimes the derived terms are so confusing, e.g. "left/right inverse function", their existence or properties.
so, which should I choose for the future studies? is there differences between fields or not?
I would say that $(f\circ g)(y) = f(g(y))$ is most commonly used and recognised in (almost?) all fields of mathematics. I highly recommend choosing this for further studies.
"Left-applied" functions, i.e. writing $(y)f = x$ when $f$ maps the element $y$ to $x$, would be the alternative. In this case one would have $(y)(f\circ g)=((y)f)g$, which is your second case.
I have never seen this in actual use, only mentioned as a possibility and used with prior explanation of notation. (One situation where I have seen something like this is when $y\in V^*$ is an element of the dual of a finite-dimensional/Hilbert vector space and $f\in V^{**}$ is an element of the double-dual. However, in this case you can identify $f$ with its canonical counterpart in $V$, and then $y$ can be seen as a function acting on $f$.)
When you have adopted the first notation (that is, $f(y)=x$), the concepts "left-inverse" and "right inverse" are not very confusing in my opinion:
A left inverse $f_l^{-1}$ of a function $f$ is one such that placing it to the left of $f$ gives back the identity: $f_l^{-1}\circ f=\mathrm{id}$.
A right inverse $f^{-1}_r$ of $f$ is one such that placing it to the right of $f$ gives back the identity: $f\circ f^{-1}_r=\mathrm{id}$.