Following the discussion on: https://www.publichealth.columbia.edu/research/population-health-methods/difference-difference-estimation If we specify the regression equation to estimate the DiD as: $$Y= β_0 + β_1*[Time] + β_3*[Time*Intervention] + β_4*[Covariates]+ε$$
instead of :
$$Y= β_0 + β_1*[Time] + β_2*[Intervention] + β_3*[Time*Intervention] + β_4*[Covariates]+ε$$
Our $β_3$ coefficient would still be yielding the DiD estimator right?
The estimator from the first equation would be biased. One of the assumptions of difference-in-difference estimations is that potential variations between the treatment and control group are accounted for and constant over time (parallel trends assumption). That's the "constant difference in outcome" in Figure 1 from your link. If you don't control for Intervention, then that constant difference assumption is no longer part of your parametric equation.
Similarly, you must control for Time because of the parallel trends assumption - both groups are expected to have evolved in the same way over time in the counterfactual that neither received a treatment.
Without either assumption, you no longer have a quasi-experimental design and an argument for the DiD estimator being an estimate of the causal effect of the treatment.
