Choosing linear regression model for dataset

When modelling a linear regression model for a data set there are three choices for assumptions about the influence of categorical data:

1: Assume there is no interaction effect. Regression slope results in $y = β_0 + ... + β_i*d$. Parallel slopes, varying y-intersections.

2: Assume there is an interaction effect. Regression slope results in $y = β_0...+ β_i(d*x)$. Same y-intercept. Different slopes depending on presence of dummy variable.

3: $y = β_0 + ... + β_i(d+d*x)$. Varying y-intercepts, and varying slopes.

Interaction effects could be discovered with an ANOVA analysis, but when should the third model be used?