Where I work, there used to be quite a bit of a confusion on which rates one should use as input to a Local Volatility Monte-Carlo simulation.
In particular there is a paper in the Journal of Computation Finance by Andersen and Ratcliffe "The Equity Option Volatility Smile: a Finite Difference Approach" which explains one should use specially tailored rates for the finite difference scheme in order to reproduce exact Bond price and exact Forward contract prices.
Code has been updated and roll-backed, people have complained around it. But nobody really made the effort to simply write clearly what's going on, or even write a unit test around it. So it was just FUD, until this paper.
In short, for log-Euler, one can use the intuitive forward drift rate: r1*t1-r0*t0 (ratio of discount factors), but for Euler, one need to use a less intuitive forward drift rate to reproduce a nearly exact forward price.