Centered Fourier transform
the "Non-Centered" is the same as the Discrete Fourier Transform

$$ X[k] \triangleq \sum\limits_{n=0}^{N-1} x[n] \, e^{-i 2 \pi \frac{nk}{N}} $$

when frequency $f = \frac{k}{N}$ is normalized to the sample rate ($f=\tfrac12$ means the Nyquist frequency). And it assumes all of $x[n]$ is for non-negative values of time.  "$n \ge 0$" is "time" in units of the sampling period.  $n=0$ means time at $t=0$.

the "Centered" is the same as the DFT with a linear phase-shift factor

$$ e^{i \pi (N-1) f} X[k] $$

when $f = \frac{k}{N}$ and assumes $x[n]$ is for instances of time that are both positive and negative.  $n=\frac{N-1}{2}$ means time at $t=0$.  if $N$ is odd, then the "time origin" is exactly the time of the sample at $x[\tfrac{N-1}{2}]$.  but if $N$ is even, the time origin (when $t=0$) is midway between adjacent samples $x[\tfrac{N}{2}-1]$ and $x[\tfrac{N}{2}]$.  if you use the DFT to compute your "Centered" Fourier transform, you probably want to remove that linear phase-shift factor by multiplying the DFT result, $X[k]$, with $e^{-i \pi (N-1) f}$ or $e^{-i \pi (N-1) k/N}$.  that will place the time origin (when $t=0$) where $n=\frac{N-1}{2}$.