Standard practice is to include the dummy variable as well. Using your equations, we could say that x represents age, dum represents sex (0 = female, 1 = male) and Y represents income. In the first equation, we would get an estimate of the effect of age (b1) on income and an estimate of the difference in effect of age on income. In the second equation, we will also get an estimate of the effect of sex on income. A major reason to include the effect of sex on income, even if we might not be at all interested in that effect, is that if there is a difference, we will get a more accurate estimate of the regression coefficient of the interaction.
If we do not include the main effect of sex (as in your first equation), the specific effect of sex on income will distort the estimate of the effect of the interaction.
I haven't heard of any advantages of leaving out the main effect unless you are certain that there is no effect (in which case it shouldn't matter if you leave it out) and you have a small sample size so that the number of parameters you can estimate is a serious limitation.