I understand that the standard difference in differences with 2 groups, 2 time periods appears as follows \begin{align*} y = \beta_0 + \beta_1Tr + \beta_2Post + \beta_3 Tr \times Post + u \end{align*} Where $Tr$ is a dummy for being in the treatment group, $Post$ is a dummy for the post treatment time period. However, I have seen models utilizing multiple time dummies, formulated as follows
\begin{align*} y = \beta_0 + \beta_1Tr + \sum_{t = 1}^T\lambda_td_t + \beta_2D + u \end{align*} where each $dt$ is a dummy for the time period (among $T$ total time periods), and \begin{align*} D = \begin{cases} 1 \text{ if in treatment group and post-treatment}\\ 0 \text{ otherwise} \end{cases} \end{align*} I am confused as to the purpose of the second formulation, if ultimately we are still interested in the effect on the interaction term. If we say, have 10 time periods, what is the difference between including all 10 time dummies vs. just taking one pre- and one post- time period? Thank you.