Going into the 1960s it seems to me that topologists saw path spaces as an advanced idea, useful in come contexts but not fundamental. So they took homotopy of maps as basically what is now called left homotopy, with today's right homotopy merely an alternate version which was only of interest insofar as it agrees with the basic definition in well behaved cases. But I may be wrong about that.
Who first distinguished left and right homotopy as equally fundamental ideas, which need not agree in every situation?
It seems likely that Quillen was the first. Hopf had long before used the fact that a homotopy between maps $f,g:S\rightarrow T$ can be regarded as map from the product $S\times[0,1]$ to $T$, or a map from $S$ to the function space $T^{[0,1]}$. All interested topologists knew this would not work for all topological spaces $T$. Steenrod's work with Eilenberg on axiomatic cohomology made him appreciate categorical structure in topology, and Whitehead Elements of Homotopy Theory p. iii says Steenrod noticed homotopy theory works far better in a context such as compactly generated spaces where "the exponential law, relating cartesian products and function spaces, [is] universally valid."
Quillen was informed by all of that work, plus the then-nascent functorial cohomology of the Grothendieck school. And he cites Lawvere's work on functorial tools in logic, so he likely knew Lawvere's use of the fact that a natural transformation between functors $f,g:S\rightarrow T$ can be regarded as functor from the product $S\times 2$ to $T$, or a functor from $S$ to the functor category $T^2$. And in the setting of categories the exponential law relating cartesian products and functor categories is universally valid and can actually be used to define either one of those from the other. So these two definitions of natural transformation are entirely equivalent. Both Quillen and Lawvere would have known that a category is like a (two dimensional) simplicial set, and a functor like a simplicial map, so that a natural transformation is like a homotopy.
Apparently Quillen was the first to put that all together by axiomatizing homotopy. using "left homotopy" based on products. and "right homotopy" based on loop spaces.