In recent years, there has been increasing interest within the computational social choice community regarding models where voters are biased towards specific behaviors or have secondary preferences. An important representative example of this approach is the model of truth bias, where voters prefer to be honest about their preferences, unless they are pivotal. This model has been demonstrated to be an effective tool in controlling the set of pure Nash equilibria in a voting game, which otherwise lacks predictive power. However, in the models that have been used thus far, the bias is binary, i.e., the final utility of a voter depends on whether he cast a truthful vote or not, independently of the type of lie. In this paper, we introduce a more robust framework, and eliminate this limitation, by investigating truth-biased voters with variable bias strength. Namely, we assume that even when voters face incentives to lie towards a better out-come, the ballot distortion from their truthful preference incurs a cost, measured by a distance function. We study various such distance-based cost functions and explore their effect on the set of Nash equilibria of the underlying game. Intuitively, one might expect that such distance metrics may induce similar behavior. To our surprise, we show that the presented metrics exhibit quite different equilibrium behavior.