Confusion between function learned and the underlying distribution

Let us assume that I am working on a dataset of black and white dog images.

Each image is of size $$28 \times 28$$.

Now, I can say that I have a sample space $$S$$ of all possible images. And $$p_{data}$$ is the probability distribution for dog images. It is easy to understand that all other images get a probability value of zero. And it is obvious that $$n(S)= 2^{28 \times 28}$$.

Now, I am going to design a generative model that sample from $$S$$ using $$p_{data}$$ rather than random sampling.

My generative model is a neural network that takes random noise (say, of length 100) and generates an image of the size $$28 \times 28$$. My function is learning a function $$f$$, which is totally different from the function $$p_{data}$$. It is because of the reason that $$f$$ is from $$R^{100}$$ to $$S$$ and $$p_{data}$$ is from $$S$$ to $$[0,1]$$.

In the literature, I often read the phrases that our generative model learned $$p_{data}$$ or our goal is to get $$p_{data}$$, etc., but in fact, they are trying to learn $$f$$, which just obeys $$p_{data}$$ while giving its output.

Am I going wrong anywhere or the usage in literature is somewhat random?

You're right! The generative model $$f$$ is not the same as the probability density (p.d.f.) function $$p_{data}$$. The kind of phrases you've referred to are to be interpreted informally. You learn $$f$$ with the hope that sampling a latent vector $$z$$ from some known distribution (from which it is easy to sample), results in $$f(z)$$ that has the probability density function $$p_{data}$$. However, merely learning $$f$$ does not give you the power to estimate what $$p_{data}(x)$$ is for some image $$x$$. Learning $$f$$ only gives you the power to sample according to $$p_{data}(\cdot)$$ (if you've learned an accurate such $$f$$).