It’s both. Fine-tuning a BERT model changes its weights, which causes the embeddings to change.
For example you might have one model which embeds a text query and another model which embeds an image. You also have a dataset of image + text captions. Training means updating the weights of those models so that the embedding of the image is close to the embedding of its corresponding caption.
I thought that fine-tuning was changing the weights in the model, not the embedding? Or did I misunderstand?