Is my transform the essence of DFT?
I think you could say that in simple terms this is the essence of the DFT. Just note that the DFT correlates against complex exponentials:

$$X[k]=\sum_{n=0}^{N-1}x[n]e^{-j2\pi nk/N}\tag{1}$$

You've just computed the (negative) imaginary part of $(1)$ (for $k=2$):

$$-X_I[k]=\sum_{n=0}^{N-1}x[n]\sin(2\pi nk/N)\tag{2}$$

For your example that's also sufficient because the real part is zero since your signal $x[n]$ is one full period of a sine wave. However, imagine you added a phase to your signal, then you would also need the real part of $(1)$. In the worst case you could choose a cosine for $x[n]$, which would give you a correlation of zero, even for the same frequency, if you just correlated with sine functions as in $(2)$.