  We prove new a posteriori error estimates for surface finite element methods
(SFEM). Surface FEM approximate solutions to PDE posed on surfaces.
Prototypical examples are elliptic PDE involving the Laplace-Beltrami operator.
Typically the surface is approximated by a polyhedral or higher-order
polynomial approximation. The resulting FEM exhibits both a geometric
consistency error due to the surface approximation and a standard Galerkin
error. A posteriori estimates for SFEM require practical access to geometric
information about the surface in order to computably bound the geometric error.
It is thus advantageous to allow for maximum flexibility in representing
surfaces in practical codes when proving a posteriori error estimates for SFEM.
However, previous a posteriori estimates using general parametric surface
representations are suboptimal by one order on $C^2$ surfaces. Proofs of error
estimates optimally reflecting the geometric error instead employ the closest
point projection, which is defined using the signed distance function. Because
the closest point projection is often unavailable or inconvenient to use
computationally, a posteriori estimates using the signed distance function have
notable practical limitations. We merge these two perspectives by assuming {\it
practical} access only to a general parametric representation of the surface,
but using the distance function as a {\it theoretical} tool. This allows us to
derive sharper geometric estimators which exhibit improved experimentally
observed decay rates when implemented in adaptive surface finite element
algorithms.
