@AndreasSteimer every reference I have come across has always defined the likelihood as a function of the parameters given fixed data. The opposite situation where probabilities are returned as a function of data given fixed parameters would simply be a probability distribution or density.

@whuber that makes perfect sense and I would gladly accept it as an answer.

One small pedantic point: that the model doesn't change with coding isn't always true; if you use a strong prior on your global intercept, then you would be restricting how much it can change and so a diference in coding would actually lead to somewhat different predictions. But this is why Bayesian modeling guidelines strongly recommend that global intercept terms are given wide priors and brms follows that advice so you shoud not run into any issues related to this in practice.

Anyway, I definitely should have been more concise here, sorry about that! I was discovering the answer as I wrote it. To summarize: contrast coding does not "mess up" the global intercept in the sense of making it take some undesirable value, but it does change its value and and its interpretation. My point about the group level intercepts is that they could be what changes instead of the global intercept, but because all statistical software forces their mean to be zero, then you're also guaranteed that will not happen and the global intercept will always be the only thing which changes.

Hmm, wait, you mention group-level intercepts and a global intercept, which I showed above, but then also mention group-level effects and global effects. Can you clarify what you mean with these terms? In my example, and in mixed models in general, the "slope" ($\beta$ here) is interpreted as the effect conditional on group membership, which one could parse as "group level effect", however it's a single identical parameter shared across all groups, so you could also say it's a "global effect".

Small pet peeve: I believe you are referring to the same coding (-0.5, 0.5) by sum, effect, deviant and deviation coding. Please try to keep the terminology consistent! I'm interested in an answer to this myself, I'll do some simulations and report back.

"Participants each get to try 10 cakes (set A) blindfolded and 10 (set B) without a blindfold." Please clarify: what is the difference between set and blindfold state? The phrasing there would make them seem to be the same, but the dataframe you show later has both blindfold states within A and B but no flavor overlap, therefore making it seem like set A/B partition flavors instead.

You mention that linear probability models are "exotic" but from my interactions with economists (I'm a statistician myself) it seems there are two camps, one of which argues that linear probability is better because it involves less assumptions and directly models the expectation, which is what one normally cares about.