As far as I know that the sun exclusively produces electron neutrinos ($\nu_e$). When the flux of solar neutrinos ($\nu_e$) is measured on the earth, a depletion is observed in the $\nu_e$ flux i.e., some $\nu_e$'s have "disappeared" in their way from the sun to the earth. As far as I know that this conclusion is drawn, by measuring only$^1$ the $\nu_e$ flux in the detectors. The explanation is that some of the neutrinos get morphed into $\nu_\mu$ and $\nu_\tau$.
But to really test this hypothesis, a deficiency in the $\nu_e$ flux is not enough. There must be an experiment where the detector must also measure the $\nu_\mu$ and $\nu_\tau$ fluxes. If adding the fluxes over all three flavors turns out to be equal to the expected flux then only we can be sure that solar neutrinos have undergone oscillation. Has that been achieved in experiments?
$^1$The experiment carried out by Davis et al at the Homestake mines detected $\nu_e$ through the inverse beta decay $\nu_e+^{37}{\rm Cl}\to e^-+^{37}{\rm Ar},$ and found that they were getting about one-third of the number of $\nu_e$ that were predicted from the solar models.
Answer
The SNO experiment was sensitive to all three flavors of neutrinos, and hence provided definitive evidence for solar neutrino oscillations. That's why half of the 2015 Nobel prize for neutrino oscillations went to the director of this experiment, rather than the many other previous experiments.
As usual, there's a bit of confusion depending on whether you talk primarily to theorists or experimentalists. Judging from the model building occurring in the 90's, by 2001 most theorists considered it a given that neutrinos had masses and hence oscillated. But the SNO experiment was the first to really check it for sure.
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