@HeteroskedasticJim I don't mind the pedantic points, quite the contrary. Also, I really appreciate that you took the time to look at my unresolved questions. :)

This last consideration was probably important at the time when the models were developed since computing power was limited and the mathematical particularities of each model were exploited to find efficient approaches. Today I don't feel it's very important as the fitting process has been largely abstracted from the end user and recent advances in Bayesian computation mean a single algorithm can be used for an immense amount of models.

As for the General Linear Model, it doesn't allow for a "link" function and directly relates the expectation of the potentially multivariate responses with the linear predictor. If this isn't clear you're not alone in your confusion. As to why these distinctions are necessary, I guess it's a matter of both historical usage and the different techniques that are used to fit each type of model.

Yes, I mean multiple outcomes. The definition put forth in McCullagh's monograph introducing GLMs specifies that the response variable must be a vector. Under the assumptions GLM works under, you can decompose a multinomial response so that each "draw" behaves a single distribution, and if you choose to represent the outcome as a numeric one (eg first option is 1, second is 2 and so on), you can produce a link function that take a single linear predictor. More explanation here.

Also note that if you do think time has some stable but not necessarily monotonic effect, you can include a squared term (in principle you can include an arbitrary amount of higher order terms but you'd need to come up with a very good explanation for that and run the risk of overfitting).

This is a question only you can answer. If you represent time as a categorical variable, you're essentially saying that you expect each measurement to exhibit a unique characteristic, unrelated (in the time scale) to previous and future measurements. If you include it as integer, then you're saying that time has a monotonic and stable effect across measurements and, importantly, also across unmeasured periods (you can simply use t*coeff to see its effect between measurements).

I've been looking for this derivation for a long time! I'm wondering, in what context did you have to develop this knowledge? Did you see this as part of a course or textbook? I kept finding references to this relationship on the internet but no one actually gave the details.

@whuber you mention "that is not the purpose of the link function". What could we say its purpose is then? I intuitively understand it "reshapes" the response space to one where unrestricted linear combinations can be taken, but this does not outright explain why we commonly choose continuous, monotonic functions (maybe it's just because they're simple choices that normally work? or are there more fundamental reasons?).

Yes, that would be appropriate.