Why are there so many fractional derivatives?

The reason is that the fractional derivative is not a local operator. The usual derivative is a local derivative in the sense that the value of the derivative at one point only depends on the value of the function in a neighborhood of that point. This is not the case for the fractional derivative and that cannot be due to some general theoretical result due to Peetre.

So the definition depends on the domain of definition of the functions under scrutiny. This is not the same definition if we are looking at functions defined on ${\bf R}$ or on $[0,1]$ or on $[0,\infty)$ and of course the derivative of say $\sin$ is not the same in these three cases. Same for the derivative of the constant function.

Fractional derivatives are a particular example of operators obtained using the functional calculus on some operator space. The result of such operation of course depends on the functional space under consideration, which itself is dictated by the context and the problems at hand.

tl;dr: there is not a best definition and the fractional derivatives do not share the nice local properties of the usual derivative, so beware.

A general framework that gives meaning to functions of differential operators that are not restricted to powers is the theory of pseudodifferential operators.

It is not at all beyond the bounds of possibility that a unifying theory of fractional differintegration will emerge, either within the theory of pseudodifferential operators or outside it. If it does, how it will relate to other functions of the differential operator than powers is not known.

Are some of (the many definitions of the fractional derivative) "better" than the others in some sense?

I'm not sure about this. As well as Riemann-Liouville, Grunwald-Letnikov, and Caputo, the others have included some that have hardly been referred to after they were announced. The hottest definition in the fractional calculus field at the moment - by number of papers that refer to it - is the Atangana-Baleanu one, which I asked about here.