If we assume a standardized treatment adoption period for all treated entities, then it simplifies things. I reproduced your first model below:
$$
y_{i,t} = \lambda_i + \tau_t + \beta (Treat^1_i \times Post_t) + \delta (Treat^2_i \times Post_t) + \eta_{i,t},
$$
where I superscripted the numerals to index the different treatments. Here, we have three exposure groups (i.e., control group, treatment group 1, treatment group 2) and two contrasts. You are comparing $Treat^1_i$ with the control group and $Treat^2_i$ with the control group in one big regression. $Post_t$ is well-defined so we can proceed in this manner. Once different entities (or groups of entities) have different adoption periods, then we need to approach this in a different way. For now, the "classical" difference-in-differences (DD) approach with a post-treatment indicator specific to all groups is appropriate. Note, you could actually run separate DD models on subsets of your data and obtain the same estimates. One subset would include all controls and $Treat^1_i$ entities—only; likewise, the other would include all controls and $Treat^2_i$ entities—only. However, I would go with one big fat regression. This post also addressed a very similar specification.
I should note a concern. Including $\lambda_i$ and $\tau_t$ is fine, but software (e.g., R) will drop three main effects due to singularities. For instance, $Treat^1_i$ and $Treat^2_i$ are collinear with the the unit fixed effects (i.e., $\lambda_i$) and will be dropped. Similarly, $Post_t$ is collinear with the time fixed effects (i.e., $\tau_t$) and will be dropped as well. Don't worry, removal of the main effects should not affect your estimates of $\beta$ and $\delta$. Either ignore the singularities in your output, or drop the fixed effects. In settings like yours where you have a well-defined exposure period, the interaction of a treatment dummy with a post-treatment indicator is all that is needed.
Where I omit the event year -1, one year prior to treatment. Also assume that both treatments occur at the same time, so k = -1, event year is the same year for each treatment. Does this yield the normal interpretation of event studies for each estimate of 𝛽 and 𝛿?
Yes. We still have the same contrasts. Reproducing your equation:
$$
y_{i,t} = \lambda_i +\tau_t + \sum\limits_{k \neq -1}Treat^1_i * \mathbb{1}\{t=k\}\beta_k + \sum\limits_{k \neq -1}Treat^2_i * \mathbb{1}\{t=k\}\delta_k + \eta_{i,t},
$$
where you now saturate your equation with time (year) dummies. Your reference is the year before treatment (i.e., $k = -1$) or whatever year you decide to omit. In this setting, your output will display a full set of unique interactions of $Treat^1_i$ with all years and a full set of unique interactions of $Treat^2_i$ with all years. One year should (or I should say will) be omitted; the year before treatment, which is the same for the two treatment groups, is a good choice. Both treatment dummies, however, will be absorbed by the unit fixed effects; again, this should not concern you.
I think intuitively it makes sense, but my confusion is arising from the fact that in this set up, there are now 2 omitted categories, so how do I ensure that each coefficient on the treatment-event year dummies is with reference to the omitted group corresponding to that particular treatment?
In the comments you indicated that treatment begins at the same time for all units, regardless of whether they are in $Treat^1_i$ or $Treat^2_i$. You do not need to omit two periods; one period will suffice. Nothing is really changing in this specification other than we included a full set of time (year) dummies.
To put this in perspective, suppose you observe 10 districts over 10 years. Two districts fall into a low intensity treatment group denoted $T_{L,i}$ and another 2 districts fall into a high intensity treatment group denoted $T_{H,i}$. The remaining 6 receive neither treatment and serve as your control group. The intervention commences midway through your time series. All treated districts adopt some intervention in the same year but the two treatment groups vary in this "categorical" level of intensity; some districts were high in their dosage and some were low. Running the latter equation, your output will display 9 district effects, 9 year effects, 9 interactions between a low intensity dummy and indicators for all years ($T_{L,i} \times \mathbb{1}_{t = k}$) and another 9 interactions between a high intensity dummy and indicators for all years ($T_{H,i} \times \mathbb{1}_{t = k}$).
The interactions represent the unique evolution of effects for each categorical treatment group, relative to the control group, before and after the intervention. You can think of the effects in the pre-treatment epoch (i.e., $k < -1$) as placebo treatments. Hopefully you do not observe the consequences of the intervention before it starts! Any strong non-zero effects in the era before treatment exposure could be interpreted as selection bias.
Again, this works well when treatment timing is well-defined for all groups.
