Why is a face-centred cubic unit cell not regarded as equivalent to a body-centred tetragonal unit cell? When studying crystal lattices, I decided to learn which combinations of unit cells and lattice systems lead to the 14 Bravais lattices. I learnt that, for example, no centred triclinic unit cell exists.

Later while searching the Internet and Stack Exchange, I found an answer saying that an end-centred cubic cell can be transformed into a (smaller) tetragonal unit cell as shown in th picture below.



Then I thought on and realised that a face centred cubic lattice can be transformed into a (smaller) body-centred tetragonal unit cell, and into an even smaller one, too.

So, why are some centred unit cells (for hexagonal, triclinic and rhombohedral lattices) considered non-existent, while others (e.g. face-centred cubic) are considered to exist even though they, too, can be reduced?
In an FCC lattice  a=b=c while in a body centered tetragonal lattice a=b but c is unconstrained. The first example you gave should be that an end centered tetragonal is equivqlent to a primative tetragonal lattice. There is no end centered cubic lattice as that would violate the minimum cubic symmetry of 3 four fold symmetry axes. however the transition from FCC to a body centered tetragonal lattice would be displacive not transformational.