Feature extraction (FE) is not the same as representation learning (RL), but they are similar and related.
You describe accurately what feature extraction typically refers to, i.e. the process of extracting (new) features from existing ones or raw data (e.g. images). For example, let's say you have a dataset associated with a car. You have only two features in your dataset: distance and velocity. However, from these two, you can extract a third feature, e.g. the acceleration. So, feature extraction can be performed with a fixed algorithm (e.g. PCA for dimensionality reduction or SIFT) or manually.
Representation learning is the collection of all techniques that extract features automatically from the data (i.e. they learn the features or representations, hence the name representation/feature learning). So, for example, a convolutional neural network trained on ImageNet can (and/or needs to) learn general features in order to solve the corresponding classification task. (Chapter 9 of the book Deep Learning by Goodfellow et al. talks more about this topic.) These features are learned from the data (and that's why CNN's are data-driven), and they can later be exploited for transfer learning (TL), i.e. TL is based on the idea that neural networks learn general representations (which is thus a synonym for features) of data that can be exploited to solve other tasks (sometimes known as downstream tasks, especially in the context of self-supervised learning).
Yoshua Bengio et al. defines representation learning as follows
learning representations of the data that make it easier to extract useful information when building classifiers or other predictors
So, RL is a subset of FE, given that RL also extracts features, but RL emphasizes the extraction of features automatically.