In general, we're referring to the distribution of the residuals in these models. The distribution of $X$ in a linear regression or ANOVA model may be highly irregular, or sampled in a sequential fashion (e.g. a stratified sample of 10 people within each age group defined as 10-19, 20-29, ...). When we fit such models, we think of the $Y$s as being conditional upon the $X$s.
Homogeneity means a lack of mean variance relationship. This is assessed with the usual residual vs fitted plot to examine whether there is a trend. It seems you've already grasped this.
These "assumptions" are overplayed in applied courses, though. The only reason why we care about the normality of residuals in such models is because, when the residuals are found to be normally distributed, we can think of the tests as being obtained from a maximum likelihood estimate of the correlation/LS slope/group means.
When residuals aren't normal, we can rely on results from the Central Limit Theorem to tell us that the sampling distribution of the mean has an approximate normal distribution for even modest sample sizes. Even when sample sizes are very small, the only problem that we have with non-normality is that there is less power to detect a difference in means. If you obtain significant results in such cases, this hardly means that such results are wrong. The basic ANOVA, t-test, and linear regression models are calibrated at all sample sizes and distributions (i.e. the test is of accurate size: 0.05 cut off for p implies that 5% spurious false positives when generating data from the null hypothesis).
That's not to say that we wouldn't care about high leverage/high influence points as being obtained from an apparently skewed distribution of residuals. We might observe one or more points whose values are very inconsistent with the spread expected in a normal distribution. About 99% of the time, I leave such points in the analysis, but carefully describe what they are and how they impact the results.