A key difference between spontaneously broken symmetries and "emergent symmetries" is that emergent symmetries are never exact while spontaneously broken symmetries are backed by exact maths although the ground state isn't invariant. In most cases, the "emergent symmetries" only emerge if some parameters are fine-tuned, and even if it is so, they are only valid within some approximation scheme. In a generic situation, one has no reason to assume that a symmetry will "emerge" if it is not present fundamentally.
When there is a reason to expect such a thing, we use special names that are linked to the reason. In particular, the most solid example of an "emergent symmetry" – and a phrase that is actually being used by actual competent researchers, unlike "emergent symmetries" – is the "accidental symmetry".
It is a symmetry such as the lepton number and baryon number that is very well, approximately conserved because the terms in the equations (or action) that would violate it exist but because of a limited choice of renormalizable terms, all such terms may be shown to be high-dimension operators i.e. non-renormalizable. So their effects are negligible at low energies even though the lepton number and baryon numbers are almost certainly violated at higher energies, by the evaporating black holes or earlier than that.
In the Standard Model, the lepton number and the baryon number are conserved at the level of the renormalizable Lagrangians simply because one can't build renormalizable, gauge-invariant, Lorentz-invariant operators out of the given fields for gauge bosons, leptons, and quarks (and the Higgs field).
Your examples of Kitaev-style models and rotational symmetry are a bit less consequential. One may say that the ground state of a physical system is "rotationally invariant". But if the whole theory isn't rotationally invariant, the invariance of the ground state is pretty much a vacuous fact and its very validity is a matter of conventions (especially about a way how the symmetry-breaking theory is embedded into a larger theory that is symmetry-preserving). One won't be able to organize the spectrum into any representations of the symmetry group because it is not a genuine symmetry commuting with the Hamiltonian. Cubic crystals behave as rotationally symmetric materials in some aspects, but they see preferred directions in many other aspects.
There isn't any reason for an emergent or accidental Lorentz symmetry. This case is even much worse than the case of the "emergent rotational symmetry". In all known examples, a huge amount of fine-tuning – potentially fine-tuning of infinitely many parameters – is needed for a fundamentally Lorentz-breaking theory to reproduce Lorentz-invariant results, even at low energies. One must realize that the "maximum speed" of all the particle species including all of their possible bound states must be tuned to the same value called $c$. For each particle species, it's at least one additional tuning. There's no reason why all these fine-tunings should conspire and work properly so no viable theory in physics can make such assumptions about "emergent symmetries".
There's no name used by experts for "emergent Lorentz symmetry" etc. because the phenomenon envisioned in this name can't physically occur. The OP made it sound that this is just a formality and one only needs to learn the "right name". But physics isn't about terminology. The first question is whether such a hypothetical mechanism occurs in Nature and the answer is essentially No. So there's nothing to invent names for.