You're right that what I described isn't what people commonly think about as embeddings (given we are more advanced now the above description), but broadly an embedding is anything (in nlp at least) that maps text into a fixed length vector. When you make embedding like this, the nice thing is that cosine similarity has an easy to understand similarity meaning: count the number of words two documents have in common (subject to some normalization constant).
Most fancy modern embedding strategies basically start with this and then proceed to build on top of it to reduce dimensions, represent words as vectors in their own right, pass this into some neural layer, etc.
Most fancy modern embedding strategies basically start with this and then proceed to build on top of it to reduce dimensions, represent words as vectors in their own right, pass this into some neural layer, etc.