The version number 2 is the only one that makes the Fourier transform both a unitary operator on $L^2$ and an algebra homomorphism from the convolution algebra in $L^1$ to the product algebra in $L^\infty $.
It is not, however, of widespread use in analysis as far as I know. From the point of view of semiclassical analysis, it amounts roughly speaking to consider Planck's constant $h $ rather than $\hslash=h/2\pi $ as the semiclassical parameter (or as the constant set to one in quantum systems). This is somewhat differing from the common practice in physics.
The Fourier transform $F(\mathbf{k})$ of the unit ball in L1 metric can be evaluated as follows$^\ast$.
Notation: $\theta(x)$ is the unit step function, $\delta(x)$ is the delta function, and ${\cal P}$ denotes the principal value of the integral.
$$F(k_1,\ldots k_n)=\int \cdots\int e^{i\mathbf{k\cdot x}}\theta\left(1-\sum_{p=1}^n|x_p|\right)dx_1\cdots dx_n$$
$$\qquad=\frac{1}{2\pi i}{\cal P}\int_{-\infty}^\infty \frac{ds}{s}\left(1-e^{-is}\right)\prod_{p=1}^n\left(\pi\delta(k_p-s)+\pi\delta(k_p+s)+\frac{2is}{s^2-k_p^2}\right)\qquad \mathbf{(1)}$$
$$\qquad =\frac{1}{2\pi i}{\cal P}\int_{-\infty}^\infty \frac{ds}{s}\left(1-e^{-is}\right)\prod_{p=1}^n\left(\frac{2is}{s^2-k_p^2}\right)$$
$$\qquad\qquad\qquad\qquad+\,\text{Re}\,\sum_{q=1}^n\left[\frac{1-e^{-ik_q}}{ik_q}\prod_{p=1,p\neq q }^n\left(\frac{2ik_q}{k_q^2-k_p^2}\right)\right].\qquad \mathbf{(2)}$$
Then finally I arrive at the result
$$F(k_1,k_2,\ldots k_n)=2\,\text{Re}\,\sum_{q=1}^n\left[\frac{1-e^{-ik_q}}{ik_q}\prod_{p=1,p\neq q }^n\left(\frac{2ik_q}{k_q^2-k_p^2}\right)\right].\qquad \mathbf{(3)}$$
I checked that for $n=2$ and $n=3$ this result agrees with the direct evaluation of the integrals (expressions given in the comment).
$^\ast$ Details of the calculation:
I first replace $\theta(1-\sum_p|x_p|)$ by $\theta(\xi-\sum_p|x_p|)$, denote the $\xi$-dependent integral by $F_\xi$, and take the derivative with respect to $\xi$, $F_\xi\mapsto F'_\xi$, so that the step function becomes the delta function $\delta(\xi-\sum_p|x_p|)$. Then I Fourier transform with respect to $\xi$, $F'_\xi\mapsto G_s$. The $n$-fold integral of $G_s$ over $x_1$, $x_2, \ldots x_n$ factorizes, so I can carry it out explicitly, using the identity $\int_0^\infty e^{ikx}\,dx=\pi\delta(k)+i{\cal P}k^{-1}$; finally I Fourier transform back $G_s$ to the $\xi$ variable, to obtain $F'_\xi$. One more integration of the $\xi$ variable, using $F_{\xi=0}=0$, and then setting $\xi=1$ gives the first integral expression (1).
For the second integral expression I may safely assume that all $k_p$'s are distinct, so products of delta functions do not contribute. I can then integrate out the single delta functions, obtaining the second integral expression (2).
The integral can then be performed by closing the contour in the lower half of the complex plane, the poles are shifted to the lower half of the complex plane, and the residue is reduced by a factor of two to account for the principal value. It then turns out that the integral equals exactly the contribution from the delta functions, so I end up with the final result (3).
Best Answer
At the risk of answering my own question, here is what I have since found:
For general $n$, formula (1) seems to occur first on p. 177 of S. Bochner, Summation of multiple Fourier series by spherical means, Trans. AMS 40 (1936) 175-207. Bochner exposes it again on pp. 73-74 of Fourier Transforms (Princeton UP 1949).
For $n=3$, Burkhardt (Trigonometrische Reihen und Integrale bis etwa 1850, Encykl. Math. Wiss. II A 12 (1916) 819-1354, page 1258) claims to find formula (3) in Poisson's Mémoire sur l'intégration de quelques équations linéaires aux différences partielles, et particulièrement de l'équation générale du mouvement des fluides élastiques, Mém. Acad. Roy. Sci. Inst. France 3 (1820) 121-176, page 134, in the form $$ \mathfrak{Sin}\,pt= \frac{pt}{2\pi}\int_0^{2\pi}\int_0^\pi\exp\{t(g\cos u+h\sin u\sin v+k\sin u\cos v)\}\sin u\,du\,dv $$ where $p=\sqrt{\smash[b]{g^2+h^2+k^2}}$, $\mathfrak{Sin}$ is a hyperbolic function, and Burkhardt is missing a factor of 2. However... I'm not able to find it on that page of Poisson. On the other hand Poisson states it as "known" in a later memoir (1831, page 558). Perhaps someone will have better luck locating the original (3) -- in Poisson or elsewhere?
Edit: Aha, the problem was simply a typo in Burkhardt. Formula (3) indeed appears in Poisson's above-cited Mémoire, but on page 174 instead of 134, in the form $$ \int\int e^{at(g\cos u+h\sin u\sin v+k\sin u\cos v)}\sin u\,du\,dv = 2\pi\frac{e^{atp}-e^{-atp}}{atp}. $$