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?

You need to search more about these terms. Resources may contain 'Linear Algebra', 'Signal & Control', 'Data Mining', 'Predictive Models',... vice versa. I explain some of them briefly here:
Interpolation: Some times we need to predict value of some data points based on another ones, that is generalizing the order we found on some observations (sampled data points) , to another one. Observed data points might be sampled points. We describe this order as mathematical functions (named basis function). This function may be a linear function; in this case we call it Linear Regression. If interpolation is based on current observed values, and aims to predict upcoming observations (assumes that current trend of data points continues on future) we call the method 'Extrapolation'.
Regression: Is the process of finding the best values for  parameters of a regression function (for example parameters of a line) in-order to fir into data points. Regression about  doesn't have any scenes about trend, it just tries to optimize the parameters of a linear function and find their best values, which fit best to data points. A polynomial regression tries to fit the best polynomial function to data points.
Classification: Some times you want to assign a category to data points. Classes are finite and are pre-known. It is like a process of mapping data points from a infinite space, into some points in a finite space. In Regression, you was trying to search for best values for a basis functions parameters, but in classification you are trying to find best values for a Mapping function. A mapping function could also be a line, a polynomial, or each function which was applicable in interpolation (and regression). The only difference is the way we use it. We use mapping function for separating data points from each other, while use basis functions in regression for finding out the order which describes them the best. Neural Networks (which are used vastly in classifications) contain several mapping matrices which maps input data described in finite properties to output finite classes.
Prediction: Could be prediction value of future data points or their class. That is a general term which can contain both Regression, and Classification. Predication can employ any mathematical approach with the assumption that the current order of events, will continue in future, or any changes will follow an order which could be known or estimated.
Modeling: Is another general term (more general one) which is the way we describe a system by a mathematical (or any thing) abstract model. The output of this process is a tool for predicting, interpolation, describing, and analyzing a system. Regression is a kind of modeling which describes the system as a linear function and could be used for prediction and describing.
