= Solution
Let $Y_{go}$ be the count for treatment
$$
g\in\{C,L,S\}
$$
(Control, LD, or SD) and outcome
$$
o\in\{B,W\}
$$
(Better or Worse), and write $\mu_{go}=\mathbb E Y_{go}$. Both fits take the six cell counts to be independent <Poisson distributions>[Poisson] variables.
The additive <log-linear model> fitted by `fit1` is
$$
Y_{go}\sim\operatorname{Pois}(\mu_{go}),
\qquad
\log\mu_{go}=\lambda+\alpha_g+\beta_o,
$$
with reference constraints $\alpha_C=0$ and $\beta_B=0$. Thus treatment changes the overall group count but not the relative frequencies of the two outcomes.
The interaction model fitted by `fit2` is
$$
Y_{go}\sim\operatorname{Pois}(\mu_{go}),
\qquad
\log\mu_{go}
=\lambda+\alpha_g+\beta_o+\gamma_{go},
$$
where, under reference coding,
$$
\gamma_{CB}=\gamma_{CW}=\gamma_{LB}=\gamma_{SB}=0
$$
and the two free interactions are $\gamma_{LW}$ and $\gamma_{SW}$. This is the <saturated log-linear model> for the $3\times2$ table.
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