  We perform a comprehensive study of the total mass distribution of the galaxy
cluster RXCJ2248 ($z=0.348$) with a set of high-precision strong lensing
models, which take advantage of extensive spectroscopic information on many
multiply lensed systems. In the effort to understand and quantify inherent
systematics in parametric strong lensing modelling, we explore a collection of
22 models where we use different samples of multiple image families,
parametrizations of the mass distribution and cosmological parameters. As input
information for the strong lensing models, we use the CLASH HST imaging data
and spectroscopic follow-up observations, carried out with the VIMOS and MUSE
spectrographs, to identify bona-fide multiple images. A total of 16 background
sources, over the redshift range $1.0-6.1$, are multiply lensed into 47 images,
24 of which are spectroscopically confirmed and belong to 10 individual
sources. The cluster total mass distribution and underlying cosmology in the
models are optimized by matching the observed positions of the multiple images
on the lens plane. We show that with a careful selection of a sample of
spectroscopically confirmed multiple images, the best-fit model reproduces
their observed positions with a rms of $0.3$ in a fixed flat $\Lambda$CDM
cosmology, whereas the lack of spectroscopic information lead to biases in the
values of the model parameters. Allowing cosmological parameters to vary
together with the cluster parameters, we find (at $68\%$ confidence level)
$\Omega_m=0.25^{+0.13}_{-0.16}$ and $w=-1.07^{+0.16}_{-0.42}$ for a flat
$\Lambda$CDM model, and $\Omega_m=0.31^{+0.12}_{-0.13}$ and
$\Omega_\Lambda=0.38^{+0.38}_{-0.27}$ for a universe with $w=-1$ and free
curvature. Using toy models mimicking the overall configuration of RXCJ2248, we
estimate the impact of the line of sight mass structure on the positional rms
to be $0.3\pm 0.1$.(ABRIDGED)
