Questions and comments:

What is "exponential" Poisson? Does that mean you have a log link?

What is "cluster-adjusted" SE? Does that mean you had multiple observations for subject or cluster, so you are controlling for between subject variance as well? If so, then your written rationale jives with my understanding as well.

As for the mystery of marginal effect, I am in your boat. I think it is just another case of how people use terms. My personal understanding is that if you are interpreting a single effect then it is marginal (say 2x2 table, numbers are on the margins of the table). But if you are adjusting or controlling for something else (especially for an interaction), now instead of having a marginal distribution, the effects are based on a joint distribution, so you have conditional effects. Let me know if I need some corrections here as well, also, is your interaction cross-level. Interacts with individual and group levels, but you are not directly controlling for clusters, just using robust SE?

P.S., Robust SEs are typically wider, and if you did not change other approaches, that everything would be the same except wider confidence intervals.