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?

No, I don't think that definition is generally accepted.  I would not regard Poisson regression as classification as the thing you are generally interested is the conditional values of a Poisson distribution that describes the distribution of the target variable for those values of the attributes.  Those parameters are generally continuous.  You might then use that to work out the most likely count, but that would be discretising the predictive distribution given by the model.
Likewise some here (e.g. Frank Harrell - see his answer to this question +1) view logistic regression purely as a probabilistic model, used to estimate a conditional probability, and not as a classification model (which is what you get by applying a threshold and discretising the continuous output of the model).  I have a lot of sympathy with this view, except that in practical applications where you need to perform that discretisation, that still impacts on the design and evaluation of the model and shouldn't be ignored.  The optimal classification is not always obtained by estimating the probability of class membership and thresholding, sometimes it is better to classify the data directly.  If that were not the case, [kernel] logistic regression would not perform worse than the Support Vector Machine, but on some applications it clearly does.
I'd probably say that a classifier is a problem where the target distribution is categorical, and the aim to to place each object into a category.
