  We discuss comparatively local versus gauged Weyl symmetry beyond Standard
Model (SM) and Einstein gravity and their geometric interpretation. The SM and
Einstein gravity admit a natural embedding in Weyl integrable geometry which is
a special limit of Weyl conformal (non-metric) geometry. The theory has a {\it
local} Weyl scale symmetry but no associated gauge boson. Unlike previous
models with such symmetry, this embedding is truly minimal i.e. with no
additional fields beyond SM and underlying geometry. This theory is compared to
a similar minimal embedding of SM and Einstein gravity in Weyl conformal
geometry (SMW) which has a full {\it gauged} scale invariance, with an
associated Weyl gauge boson. At large field values, both theories give
realistic, Starobinsky-Higgs like inflation. The broken phase of the current
model is the decoupling limit of the massive Weyl gauge boson of the broken
phase of SMW, while the local scale symmetry of the current model is part of
the larger gauged scale symmetry of SMW. Hence, the current theory has a gauge
embedding in SMW. Unlike in the SMW, we note that in models with local scale
symmetry the associated current is trivial, which is a concern for the physical
meaning of this symmetry. Therefore, the SMW is a more fundamental UV
completion of SM in a full gauge theory of scale invariance that generates
Einstein gravity in the (spontaneously) broken phase, as an effective theory.
