In the context of self-supervised learning (which is also used in NLP), a downstream task is the task that you actually want to solve. This definition makes sense if you're familiar with transfer learning or self-supervised learning, which are also used for NLP. In particular, in transfer learning, you first pre-train a model with some "general" dataset (e.g. ImageNet), which does not represent the task that you want to solve, but allows the model to learn some "general" features. Then you fine-tune this pre-trained model on the dataset that represents the actual problem that you want to solve. This latter task/problem is what would be called, in the context of self-supervised learning, a downstream task. In this answer, I mention these downstream tasks.
In the same book that you quote, the author also writes (section 14.6.2 Extrinsic evaluations, p. 339 of the book)
Word representations contribute to downstream tasks like sequence labeling and document classification by enabling generalization across words. The use of distributed representations as features is a form of semi-supervised learning, in which performance on a supervised learning problem is augmented by learning distributed representations from unlabeled data (Miller et al., 2004; Koo et al., 2008; Turian et al., 2010). These pre-trained word representations can be used as features in a linear prediction model, or as the input layer in a neural network, such as a Bi-LSTM tagging model (§ 7.6). Word representations can be evaluated by the performance of the downstream systems that consume them: for example, GloVe embeddings are convincingly better than Latent Semantic Analysis as features in the downstream task of named entity recognition (Pennington et al., 2014). Unfortunately, extrinsic and intrinsic evaluations do not always point in the same direction, and the best word representations for one downstream task may perform poorly on another task (Schnabel et al., 2015).
When word representations are updated from labeled data in the downstream task, they are said to be fine-tuned.
So, to me, after having read this section of the book, it seems that the author is using the term "downstream task" as it's used in self-supervised learning. Examples of downstream tasks are thus
- sequence labeling
- documentation classification
- named entity recognition
Tasks like training a model to learn word embeddings are not downstream tasks, because these tasks are not really the ultimate tasks that you want to solve, but they are solved in order to solve other tasks (i.e. the downstream tasks).