We study two simple modifications of self-avoiding polygons (SAPs). Osculating polygons (OP) are a super-set in which we allow the perimeter of the polygon to touch at a vertex. Neighbour-avoiding polygons (NAP) are only allowed to have nearest-neighbour vertices provided these are joined by the associated edge and thus form a sub-set of SAPs. We use the finite lattice method to count the number of OP and NAP on the square lattice. We also calculate their radius of gyration and the first area-weighted moment. Analysis of the series confirms exact predictions for the critical exponents and the universality of various amplitude combinations. For both cases we have found exact solutions for the number of convex and almost-convex polygons.