As others have noted, a power series is a series $\sum_{n=1}^{\infty} a_n x^n$ (or sometimes with $x$ translated by some $x_0$, to become $(x - x_0)$). Normally
when one says Taylor series, one means the Taylor series of some particular smooth function $f$. (So in mathematical speech, one wouldn't usually say "consider a Taylor series". You might say "consider a power series", or
"consider the Taylor series of the function $f$". At least, this is my experience.)
One complication in making too much of a distinction is that any power series (say with real coefficients) is the Taylor series of a smooth function (this is a theorem of Borel). So the distinction is more terminological than logical.
Added: Borel's theorem is discussed here.
The index notation just means on the derivative just means evaluated at $x_i$,
$$
u(x) = u(x_i) + \left.\frac{du}{dx}\right|_{x = x_i}(x - x_i) + \frac{1}{2!}\left.\frac{d^2u}{dx^2}\right|_{x = x_i}(x - x_i)^2 + \cdots
$$
Forward
Take $x = x_{i + 1}$
$$
u(x_{i +1}) \approx u(x_i) + \left.\frac{du}{dx}\right|_{x_i}(x_{i+1} - x_{i}) + \frac{1}{2}\left.\frac{d^2u}{dx^2}\right|_{x_i}(x_{i+1} - x_{i})^2
+ \cdots \tag{1}
$$
up to first order in $\Delta x = (x_{i+1}-x_i)$ this becomes
$$
\left.\frac{du}{dx}\right|_{x_i} \approx \frac{u(x_{i +1}) - u(x_{i})}{\Delta x}
$$
Backward
Same deal as before, just use the point $x_{i-1}$ instead
$$
u(x_{i -1}) \approx u(x_i) + \left.\frac{du}{dx}\right|_{x_i}(x_{i-1} - x_{i}) + \frac{1}{2}\left.\frac{d^2u}{dx^2}\right|_{x_i}(x_{i-1} - x_{i})^2
+ \cdots \tag{2}
$$
truncating at first order and solving for the derivative:
$$
\left.\frac{du}{dx}\right|_{x_i} \approx \frac{u(x_{i }) - u(x_{i-1})}{\Delta x}
$$
Central
If you subtract Eq (2) from Eq (1) you get
$$
\left.\frac{du}{dx}\right|_{x_i} \approx \frac{u(x_{i+ 1}) - u(x_{i-1})}{2\Delta x}
$$
Best Answer
Both of them are the same. The second Taylor expansion that you have written is the Taylor expansion of $f(x)$ about the point $x=a$.
So in the second Taylor expansion, put $x-a=\Delta x$. See what happens.
Hope this helps you.