Confusion over terminology for time series classification

This probably is due to my lack of a reference textbook for statistical modeling of time series, anyway I'm not sure which terms we use to distinguish between two different time series classification problems.

Problem 1

We are given one time series (as in this case, which however seems to refer to regression, not classification), possibly with multiple predictors, but one categorical response. We split the data set in sequential train/test sets, with the train test preceding the test set in time, we train/fit our model on subsequences from the train set and we test/validate on the test set. Finally, we apply the model to new, unseen values of this time series. A (BAD) example would be predicting whether a certain stock, say, BIT: GOOGL, will go up or down at the end of a given day, given past values and other predictors different from past values. Of course, we could predict multiple future values at the same time, i.e., tomorrow's direction, the direction on the day after tomorrow and so on, with increasingly large prediction errors.

Problem 2

Here we don't classify one (or multiple) future samples at a time. Here we classify an entire time series. In practice we have a training set consisting of multiple time series, possibly multivariate ones: for example, clinical parameter histories (blood pressure, temperature, pulse oximeter, ECG, inspired and expired oxygen, etc.) for different individuals, as well as a category for the whole time series, not the single samples. It sounds like time series clustering, but it's supervised, instead than unsupervised. At prediction time, we get a new, complete time series (or at least a long subsequence from it), but not the label, and we must estimate the most likely label. With respect to Problem 1, Problem 2 looks closer to the classical classification problems for cross-sectional data, which we would approach with tools such as logistic regression. A peculiarity of this kind of problem for time series data is that the observations (time series) may have different lengths (you can't talk of an observation "length" in a cross-sectional setting).

1. Do you know if there are different terms for the two problems?
2. If "time series classification" is routinely used in both cases (thus leading to ambiguity), can you at least tell me to which of the two problems "time series classification" usually refers (I guess Problem 1)? This way, when people use the term in presentations, conversations, questions on Cross Validated :-) etc., I know what to expect. This applies to myself, of course, so that when I use the term here on in a conversation, I can be sure I'm not leading my interlocutor astray.