Standard model doesn't predict that neutrinos are massless. It's a "Model", where initially neutrinos are considered massless, because no mass was observed.
The way we know, now, that neutrinos have masses, is through the mixing between the different neutrino types, through a matrix called the PMNS matrix (similar to the CKM matrix for quarks). This mixing explains the number of neutrinos that come from the sun, and sets a lower limit for the masses. But no one has done a real measurement for the masses of neutrinos... dealing with neutrinos is really hard.
You're completely correct: it's perfectly allowed to have both Dirac and Majorana mass terms. However, the presence of a Majorana mass term (whether or not a Dirac mass term is present) implies the violation of lepton number. When people say they're testing for whether a neutrino is Majorana, they just mean that they're looking for such violations. For a nice review of some simple neutrino mass models, phrased in the same terms that you used, see the relevant chapter in Burgess and Moore, The Standard Model.
I don't think this necessarily is sloppy language. I think that in condensed matter, whether a fermion is Majorana or not is a sharply defined, important thing. However, in particle physics, when we say that a particle is a Blah fermion (where Blah could be Weyl, Majorana, or Dirac), we mean that we have in mind a description for that particle in terms of Blah fermion fields.
For example, a given massless neutrino state could be created by a left-chiral Weyl field, a right-chiral Weyl field, or a Majorana field. None of this affects the physics; the fields are just a bookkeeping tool that help us write down interactions for the particles. As a more extreme example, Burgess and Moore go further and describe all the fermions in the Standard Model as Majorana fields (i.e. the electron corresponds to two separate Majorana fields, but with their Majorana mass terms each set to zero), solely because this allows them to use 4-component spinors and the associated computational tools.
Historically, the distinction between Weyl, Dirac, and Majorana fields was based on the fields' Lorentz transformation properties. However, these days this is becoming less important, so the same words are repurposed. In condensed matter, the words' original meanings can't matter because there's no Lorentz symmetry, so they seem to be used to denote properties of the spectrum, or of the (anti-)commutation relations describing the system. And in particle physics, the original meanings are less important in neutrino physics for the reasons I gave above, so they are adapted to pin down the only physical thing that varies between the possibilities -- namely, whether the particle number is conserved.
Best Answer
Your question is addressed in this paper. The Standard Model as is can accomodate massive neutrinos but if the neutrinos have a mass, and no right handed neutrinos are added, the model becomes non-renormalisable. Adding right handed neutrinos fixes this.
The Standard Model doesn't make any predictions of neutrino mass, but then it doesn't predict any of the fermion masses. The masses of leptons and quarks are input parameters.