  Classes of two-nucleon ($2N$) contact interactions are developed in
configuration space at leading order (LO), next-to-leading order (NLO), and
next-to-next-to-next-to-leading order (N3LO) by fitting the experimental
singlet $np$ scattering length and deuteron binding energy at LO, and $np$ and
$pp$ scattering data in the laboratory-energy range 0--15 MeV at NLO and 0--25
MeV at N3LO. These interactions are regularized by including two Gaussian
cutoffs, one for $T\,$=$\,0$ and the other for $T\,$=$\,1$ channels. The
cutoffs are taken to vary in the ranges $R_0\,$=$(1.5$--2.3) fm and
$R_1\,$=$(1.5$--3.0) fm. The 780 (1,100) data points up to 15 (25) MeV energy,
primarily differential cross sections, are fitted by the NLO (N3LO) models with
a $\chi^2$/datum about 1.7 or less (well below 1.5), when harder cutoff values
are adopted. As a first application, we report results for the binding energies
of nuclei with mass numbers $A\,$=$\,3$--6 and 16 obtained with selected LO and
NLO $2N$ models both by themselves as well as in combination with a LO
three-nucleon ($3N$) contact interaction. The latter is characterized by a
single low-energy constant that is fixed to reproduce the experimental $^3$H
binding energy. The inclusion of the $3N$ interaction largely removes the
sensitivity to cutoff variations in the few-nucleon systems and leads to
predictions for the $^3$He and $^4$He binding energies that cluster around 7.8
MeV and 30 MeV, respectively. However, in $^{16}$O this cutoff sensitivity
remains rather strong. Finally, predictions at LO only are also reported for
medium-mass nuclei with $A\,$=$\,40$, 48, and 90.
