  Among all galactic microlensing events, those involving a passage of the
observed source star over the caustic created by a binary lens are particularly
useful in providing information about stellar atmospheres, the dynamics of
stellar populations in our own and neighbouring galaxies, and the statistical
properties of stellar and sub-stellar binaries. This paper presents a
comprehensive guide for modelling and interpreting the lightcurves obtained in
events involving fold-caustic crossings. A new general, consistent, and optimal
choice of parameters provides a deep understanding of the involved features,
avoids numerical difficulties and minimizes correlations between model
parameters. While the photometric data of a microlensing event around a caustic
crossing itself do not provide constraints on the characteristics of the
underlying binary lens and does not allow predictions of the behaviour of other
regions of the lightcurve, vital constraints can be obtained in an efficient
way if these are combined with a few simple characteristics of data outside the
caustic crossings. A corresponding algorithm containing some improvements over
an earlier approach which takes into account multi-site observations is
presented and discussed in detail together with the arising parameter
constraints paying special attention to the role of source and background
fluxes.
