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?

A reasonable standard definition of classification would be that the $Y$ value is of nominal scale level, i.e., that order and numerical differences are not meaningful (or at least not of interest). Models with count or ordinal responses are widely referred to as regression, e.g., Poisson regression, ordinal regression.
Note that the comment by Dikran Marsupial is correct that some models with nominal scaled outcome are also referred to as regression ((multinomial) logistic regression). My interpretation of this would be that the term regression is also used to refer to a class of models that grew out of historically quantitative regression models. I think it makes still sense (and would probably be standard handling of the term) to also refer to these as classification methods, and certainly to the problems solved by them as classification problems. This handling of terms would have regression and classification as not necessarily mutually exclusive. Probably regression is more ambiguous than classification, as it can be used for methods for quantitative or at least ordinal responses (and insofar be distinct from classification), but also for certain model classes that may also have versions for classification problems.
