How do I replace one tag with another, while leaving the rest of the asset's tags intact?

While the Cloudinary Upload API features a replace_tag method, this replaces all the tags for the assets in question with the tags specified in the API call. 

If you would like to simply replace one tag with another instead, there is no way to do that in one step but the two-step process is straightforward: make a call to remove one tag, and make a separate call to add the new one.

For example using the Node.js SDK:

cloudinary.v2.uploader.remove_tag('tag_to_remove', 'public_id');
cloudinary.v2.uploader.add_tag('tag_to_add', 'public_id');

Please refer to the Upload API documentation for more information on methods involved in tag management.