I was told regarding Poisson that we can't interpret the constant once there's an interaction - because an interaction term doesn't have a meaningful "baseline".
Don't know why you were told that. Perhaps it had to do with the various ways to code categorical predictors in regression models. As the different coding schemes translate the categories into predictor $x_i$ values in different ways, all regression coefficients can require some care during interpretation depending on the coding. Interaction terms can be very tricky to think about with some codings.
With the dummy or treatment coding you have in mind (the default in R), however, your sense is correct for all (generalized) linear models, including Poisson. Then for a categorical variable the reference level isn't directly coded, while all the other levels appear as mutually exclusive separate predictors having values of $0$ except for a value of $1$ for cases having that level. When all of those non-reference levels for a case have values of $0$, the case necessarily has the reference level. So you can think about $x=0$ as representing the reference level of a categorical predictor.
With that coding, the intercept $\beta_0$ is the value of the linear predictor* for a baseline condition when all categorical predictors are at reference levels and all continuous predictors have values of 0. The other $\beta$ coefficients then represent differences from that baseline, with interaction coefficients being the extra difference beyond what you would predict from the individual coefficients of the predictors involved in the interaction. The "baseline" for a two-way interaction is when either of the predictors is at baseline or has a value of 0, so that the predicted value comes just from the intercept and the individual non-interaction coefficients.
Confusion can arise if you change the reference level of a categorical predictor variable or re-center a continuous predictor, as that necessarily changes not only its own set of coefficients but also the coefficients of all predictors (categorical or continuous) interacting with it. But if you use your dummy/treatment coding, you can with some perseverance think things through.
*In a generalized linear model, the linear predictor is further transformed via the link function into a log-odds, counts, etc.