Removal and distance modeling are two common methods to adjust counts for imperfect detection in point-count surveys. Several recent articles have formulated models to combine them into a distance-removal framework. We observe that these models fall into two groups building from different assumptions about the joint distribution of observed distances and first times to detection. One approach assumes the joint distribution results from a Poisson process (PP). The other assumes an independent joint (IJ) distribution with its joint density being the product of its marginal densities. We compose an IJ+PP model that more flexibly models the joint distribution and accommodates both existing approaches as special cases. The IJ+PP model matches the bias and coverage of the true model for data simulated from either PP or IJ models. In contrast, PP models underestimate abundance from IJ simulations, while IJ models overestimate abundance from PP simulations. We apply all three models to surveys of golden-crowned sparrows in Alaska. Only the IJ+PP model reasonably fits the joint distribution of observed distances and first times to detection. Model choice affects estimates of abundance and detection but has little impact on the magnitude of estimated covariate effects on availability and perceptibility.