you are missing the discrete time fourier transform, which is the dual of your point (2).
Together, these eight domains can be arranged on a commutative cube, all of whose edges have beautiful meanings: sampling, interpolation, etc. The fact that the cube is commutative gives a lot of theorems relating each transform and these operations.
The point being that by your definition of not, DFT is also NOT a Fourier transform. It's only a "transform" in that the two domains are similar (identical, mod complexity). That's quite confusing to engineers who only care about the results. Computationally, the first two are strictly speaking impossible, but effectively, your data come in being modeled as finite real-valued arrays that represent function values, (close enough to "bounded real domain") and need to come out as finite real-valued arrays, represented integer-indexed frequencies, and a DFT on your data will operationally give you that. This is much more realistically described as a discretized pseudo-fourier series.
More precisely take a set of points, do a DFT, then make it infinite by making the rest of the terms 0. You now have the Fourier series of a periodic function that takes hits all of those points. Furthermore it is the "smallest" possible such periodic function in the L2 norm on periodic functions.
> The discrete fourier transform is actually finite-dimensional linear algebra.
Not only that, it's the most interesting kind of linear algebra: it shows that, though every (finite-dimensional if you don't like choice) vector space has a basis—so that we can pretend that all n-dimensional vector spaces are the same for a fixed n—the simple act of changing that basis (which is all the DFT is doing) can have a profound impact.
No, this is not what people mean when they say "Fourier series" which is something like "the Fourier series expansion of a periodic function".
It's not about whether the values are continuous or discrete, but instead about the domain over which the values are defined. Fourier series maps a periodic function over a continuous domain into a countably infinite number of coefficients.
