Cessium World terrain is a ion hosted asset where Bing Maps is an external asset. These also have different responses and different ways to access the data. The Bing URL is a base URL and needs to have additional information added (see BING documentation)
If you are not familar with the APIs or formats, the easiest way to access the data is to use CesiumJS which already has the loading logic coded. You can also you use that as reference if you feel you need to implement your own version.
Please let us know if this answers questions, or if you have more.
Yes. In the API documentation you will see these two button that you can use to toggle between the different responses for the Cesium hosted assets and external assets. It can be easy to miss so I wanted to make you were able to see the documentation for the external asset data as well.