Why can a face-centered cubic lattice not be redrawn as a body-centered tetragonal lattice?
OK, let's get it straight. A lattice is just a periodic set of points. For the sake of brevity, I'll talk about 2D lattices; trust me, with 3D it is pretty much the same, only more complicated.

Initially the lattices were just like that: periodic patterns of one infinitely repeating parallelogram, called the unit cell. Other than that, they were without form, and void; and darkness was upon the face of the deep. The very word "centered" was not invented yet.

Then people started to categorize lattices by symmetry.


First of all, there are those with $90^\circ$ rotational, or 4-fold symmetry (we may call them "tetragonal", or maybe "square"), and those with 6-fold symmetry (we may call them "hexagonal"). We put them aside and look deeper at the rest.


You see that the first one of these has no symmetry to speak about (other than the $180^\circ$ rotational symmetry, which all 2D lattices have by birthright). The link above calls such lattices "oblique"; I don't like the word, but whatever. The second one has mirror symmetry with respect to the lines running along the sides of the cell; this is an example of so-called "rectangular" lattice. And the third one has mirror lines along the diagonals of the cell. This just does not sit well with us; we'd rather have them along the sides. (We will meet these diagonal mirror lines later anyway, when we'll deal with the tetragonal and hexagonal lattices; but that's another story.) So we select another unit cell, twice bigger in area than the original, and call it centered.


See, the lattice (i.e., the arrangement of blue dots) is still the same; we couldn't and didn't make it more symmetric, or less symmetric, or in any slightest way different, for that matter. We just started to think of it in a different way.

And so the centered rectangular cell entered the list of 2D Bravais lattices. See, it did not get there because it can't be decomposed into some primitive cell (it surely can, as any centered cell can), or because other cells can't be centered (they surely can, only it would make no sense). It is there because we wanted our symmetry elements to be aligned in a certain way.

Like I said, in 3D we have the same story, only somewhat more complicated. Indeed, the face-centered cubic cell can be thought of as body-centered tetragonal (and also as some primitive non-centered cell, too); it is just that the cubic cell has 4-fold rotation axes running along all sides, and in a tetragonal cell some of them would run diagonally, which is not considered comme il faut.

My apologies to those who think they'd be better off with the diagonal mirrors and without centering. Anyway, the convention has been in use for well over a century now, so maybe it's not that bad after all.