classical mechanics - Why are there two definitions for the functional derivative?

I have seen two definitions for the functional derivative. Why are there two definitions?

1. In Goldstein's Classical mechanics 3rd edition page 574 eq. (13.63), and also in a Student's Guide to Lagrangians and Hamiltonans by Patrick Hamill on page 55 eq. (2.10), the functional derivative of a function $\Phi(y,y',x)$, where $y = y(x)$, is given by $$\frac{\delta \Phi}{\delta y} = \frac{\partial \Phi}{\partial y}-\frac{d}{dx} \frac{\partial \Phi}{\partial y'} .$$


2. The second definition of a functional derivative is given by $$\frac{\delta F[y(x)]}{\delta y(x')} = \lim_{\varepsilon \rightarrow 0}\frac{1}{\varepsilon} (F[y(x) + \varepsilon \delta(x-x')]-F[y(x)]).$$ This definition is found on wikipedia and is used in QFT. This definition tells us that for the functional $$S[y(x)] = \int \Phi(y,y',x)dx$$ where $y =y(x)$, the functional derivative is given by $$\frac{\delta S}{\delta y} = \frac{\partial \Phi}{\partial y}-\frac{d}{dx} \frac{\partial \Phi}{\partial y'}$$

One definition is in terms of a function and the other in terms of a functional, but both give the same output?