Regression vs. Classification: Is there a clear, generally accepted definition? As a mathematician/economist, I am not trained to think in classification and regression tasks. This is why I wonder: is there a clear, widely accepted definition of regression and classification problems?
E.g., this paper says that When $Y$ has a finite number of states we
refer to the task as classification. Otherwise we refer
to the task as regression.
Does this mean that count data with many counts are a classification problem (e.g., Poisson regression). Or if we model life expectancy, is this generally supposed to be a classification task?
Is this definition generally accepted?

Definitions are overrated. They are just words which someone claims are synonymous to other words. They may be useful in special situations, e.g. to help a novice get a grasp of a concept, or to ensure that experts, when communicating with each other, know precisely what they are talking about. But, standing alone, who cares?
That said, I believe all machine learners and a good deal of statisticians wouldn't object and would understand what you mean if you say "classification" for a task where your dependent variable is nominal (synonym: categorical) in Stevens' typology. Most machine learners will associate regression with numerical dependent variables, but some statisticians would say that "everything is regression", and classification a step---obviously outside of "everything"---you apply after performing some kind of regression (e.g. logistic regression) for the purpose of decision-making.
For dependent variables which don't fit these two categories, the things get murky. Ordinal regression lies somewhere between classical (numerical) regression and classification. Predicting counts is, in my opinion, clearly a regression (Poisson, binomial etc.), and not classification.
