Previous simulations of percolation on correlated square and cubic lattices [Phys. Rev. E 56, 6586 (1997)] have been extended to all of the common two-dimensional lattices, including triangular, square 1-2, honeycomb, and kagome. Simulations were performed on lattices of up to 1024x1024 sites. The results are independent of lattice size except, possibly, for a weak dependence at large correlation lengths. As in the previous studies, all results can be fit by a Gaussian function of the correlation length w, p(c)=p(infinity)(c)+(p(0)(c)-p(infinity)(c))e(-alpha w(2)). However, there is some evidence that this fit is not theoretically significant. For the self-matching triangular and the matching square and square 1-2 lattices, the percolation thresholds satisfy the Sykes-Essam relation p(c)(L)+p(c)(L*)=1.