How to specify a Diff-In-Diff Regression with multiple time periods?
I am a bit confused because you said there are two treatments. I presume that one "treatment" group is the control group and the other is the treatment group. (I don't think you have a separate control group.) I am also confused because you said the treatment is identical until round 3 but then the treatment begins in round 2. I presume that there is no treatment in rounds 1 and 2, and the treatment group is treated differently thereafter in rounds 3, 4 and 5.

Your $Roundn$ dummy variables for $n=2,\ldots 5$ make sure that you handle the "round effects" (for the control group) properly in a nonparametric way. This looks fine to me. If you include only the $After$ dummy, it means that there is no trend within each of the "before" and "after" periods. You would not want that.

A single $round$ variable means that there is a linear trend in $Effort$ in the control group. You could try that, but I would wonder where the linearity belief comes from. Also, you lose only 3 more degrees of freedom by including the round dummies comparing to the linear trend model. That's not a big deal unless you have a really small sample. I would be happy with the full round dummies.

Your model assumes that the treatment effects (measured by diff-in-diff) are identical in all rounds 3, 4 and 5 (because you have only one interaction term). If you believe it is true, that's fine. If you believe otherwise, you can include three interaction terms $Treatment * Round3$, $Treatment * Round4$ and $Treatment * Round5$ instead of the single interaction term. If you want, you can test if those treatment effects are identical across rounds.