Are we giving this description the status of reality? I mean is that since we describe them in this way in the SM we conclude that there must be 3 of them? [...] just one particle that interacts with an electron, a muon or a tauon with a certain probability. [...] I know they're not massless, I'm trying to understand their difference without resorting to mass since the flavours are usually introduced before you talk about their mass.
You are right that neutrino masses are "theoretical" constructs to best explain the data and that at different times the physics community becomes invested in disparate expectations of them. These expectations are incorporated in the SM, since its basic principles are sufficiently robust/permissive to accommodate several scenarios. But the WP article I sent you to explains all that in painful detail:
From 1956 to 1962, experimental evidence supported the tentative expectation there is but one neutrino, produced in association with electrons, and converting to such in a detector.
In 1962, the Lederman, Schwartz & Steinberger experiment, (Nobel prize 1988) demonstrated that the neutrino produced in association with the μ (in π decays) is essentially different and is detected in association with μs, not es, strictly. The expectation then, experimentally, surviving the discovery of the τ and nicely fitting in, was that each charged lepton is associated with a massless "flavored" neutrino characteristically, if not uniquely, associated with it. This was a pretty good picture, and the 0-th approximation of all thinking on massless neutrinos for quite a while.
(In 1998, however, it was finally discovered that, for sufficiently large L/E, these flavored neutrinos actually did couple to the "wrong" lepton upon detection, the above "flavored" neutrinos were a mirage (high energy approximation picture), and the "real" propagating neutrinos were, in fact, mass eigenstates of which the above "flavored neutrinos" were mixtures. But this is only today's best picture spatchcocked into the SM. Minds stay open and theorists speculate, somewhat hysterically.)
So, yes, a truckful of high energy experiments cannot be explained by just one neutrino, with or without mass considerations, and the simplest SM tentative "status of reality" (as you put it) summary of them is the present picture you see in charts and books and talks. One or two "popsicle graphs" won't do. You need at least three. But the answer to your question was already settled back in 1962: there has to be more than one type of neutrino.