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?

Classification denotes an action.  It's what you do with the result of an analysis in which there is one or more outcome variables and one or more input (predictor; covariate) variables.  If there is a single outcome variable, the discreteness of the variable does not matter.  For example, binary logistic regression is for binary Y and is a direct (continuous) probability model that was not intended to be used for classification.  The action of classification involves making choices and use of decision rules.  In most cases it represents a premature decision made by an analyst who is not blessed with knowledge about the consequence of the decision (i.e., does not possess the utility/loss/cost function needed to make a good decision).
One can use any predictive method to do classification even if that was not the intent of the method.   For example one can use arbitrary thresholds on predicted values to do classification from ordinary regression for continuous Y, or ordinal or binary regression for ordered or binary Y.
Many in machine learning think of classification as a good default mode; it is not, as detailed in my blog post.  Among other things, classification hides close calls and lulls users into making decisions at the boundaries (e.g., when a predicted probability is 0.5001) when a better approach would be "get more data first".
Most of the time when you see classifier used in a sentence the correct term is prediction when the output is considered to be continuous.
