Probabilistically, you can give a perfectly clear meaning to many fractional derivatives.
I will look at definitions of fractional\nonlocal derivatives that are Markovian generators of stochastic processes with jumps. I hope to convince the reader that
- Different definitions arise naturally,
- there is a clear interpretation of many properties (like nonlocality or killing/not-killing constants), and
- generalizations are natural and meaningful for applications.
It is useful to look at the most simple stochastic jump process and its corresponding generator. Take a Markov chain $P=\{p_{i,j}\}_{i,j\in \text{State space}}$ (which is intrinsically jumpy) and write out its generator
$$
\mathcal G f(x):=(P-I)f(x)=\sum_{y\in\text{ State space}}(f(y)-f(x))p_{x,y},\quad x\in\text{ State space}.
$$
Here the intuition is clear: the infinitesimal jump (working with unit time in this case) from $x$ to $y$ is assigned intensity/probability $p_{x,y}$. The operator $\mathcal G$ is non-local. If we modify the process (impose boundary conditions), say by forcing the process to be absorbed at $a\in\text{ State space}$ once it tries to jump to a state $y\notin \Omega\subset \text{State space},$ we obtain a new generator
$$
\mathcal G^{\text{abs}} f(x):=(P^{\text{abs}}-I)f(x)=\sum_{y\in\Omega}(f(y)-f(x))p_{x,y}+(f(a)-f(x))\sum_{y\notin\Omega}p_{x,y},\quad x\in\Omega.
$$
If we instead decide to kill it (by testing against functions with $f(a)=0$, for example), the new generator will be
$$
\mathcal G^{\text{kill}} f(x):=(P^{\text{kill}}-I)f(x)=\sum_{y\in\Omega}(f(y)-f(x))p_{x,y}-f(x)\sum_{y\notin\Omega}p_{x,y},\quad x\in\Omega.
$$
So from one single process we can obtain many different generators/fractional derivative (as mentioned in a comment above, the boundary conditions are reflected in the representation of the operator away from the boundary due to the non-locality of $\mathcal G$).
Let us now move to the Riemann-Liouville and Caputo derivatives of order $\beta\in(0,1)$. Consider the three fractional derivatives for $x<a$ \begin{align} D^{\beta}_{\infty}f(x)&:= \int_0^{\infty}(f(x+y)-f(x))\nu(y)dy, \\ ^{C}D^{\beta}_a f(x):&= \int_0^{a-x}(f(x+y)-f(x))\nu(y)dy &+(f(a)-f(x))\int_{a-x}^\infty\nu(y)dy,\\ ^{RL}D^{\beta}_af(x)&:= \int_0^{a-x}(f(x+y)-f(x))\nu(y)dy &-f(x)\int_{a-x}^\infty\nu(y)dy, \end{align} where $\nu(y):=\frac{-\Gamma(-\beta)^{-1}}{y^{1+\beta}}$. Similarly as for the Markov chain above: the operator $D^{\beta}_{\infty}$ is the generator of a $\beta$-stable subordinator $X^\beta(s)$, the operator $^{C}D^{\beta}_a$ is the generator of a $\beta$-stable subordinator $X^\beta(s)$ absorbed at $\{a\}$ on the first attempt to jump outside $\Omega:=(-\infty,a)$, and the operator $^{RL}D^{\beta}_a$ is the generator of a $\beta$-stable subordinator $X^\beta(s)$ killed on the first attempt to jump outside $\Omega:=(-\infty,a)$. Integrating by parts we can rewrite the three operators above in their Riemann-Liouville integral representation, namely \begin{align} D^{\beta}_{\infty}f(x)&= \int_x^{\infty}f'(y)\frac{(y-x)^{-\beta}}{\Gamma(1-\beta)}dy \\ ^{C}D^{\beta}_a f(x)&= \int_x^{a}f'(y)\frac{(y-x)^{-\beta}}{\Gamma(1-\beta)}dy,\\ ^{RL}D^{\beta}_af(x)&= \frac{d}{dx}\int_x^{a}f(y)\frac{(y-x)^{-\beta}}{\Gamma(1-\beta)}dy, \end{align} where the last two operators are your standard definitions of Caputo and Riemann-Liouvile derivatives (right and left versions will correspond to the processes $X^\beta(s)$ and $-X^{\beta}(s)$ respectively). We can now say that the Caputo derivative $^{C}D^{\beta}_a$ (Riemann-Liouville derivative $^{RL}D^{\beta}_a$) kills (does not kill) constants as it is the generator of a process (killed process). Again you can see that (naturally) $^{C}D^{\beta}_a$ and $^{RL}D^{\beta}_a$ contain boundary information in their representation away from the boundary (in sharp difference with local differential operators). Some references: Caputo, Riemann-Liouville, and Grünwald-Leitnikov derivatives from a stochastic point of view in this book. Reflecting boundary conditions and other options for Caputo derivatives of order $\beta\in(1,2)$ here and here.
By substituting a general Lévy measure $\nu(x,dy)$ in the formulas above (generalizing fractional derivatives), many meaningful stochastic processes and their versions on a bounded domain can be studied through their generators (see book, article ). Similar arguments can be carried over for some fractional Laplacians (see this book for example).
