If I create a custom material and set the environment texture, it works as expected in Render mode That is, it acts like an environment being reflected off the object and stays relative to the viewport when the viewport is panned or rotated. But in Raytraced mode, it behaves like a diffuse texture. In other words, it stays relative to the object when the viewport is panned or rotated.
On the left is a standard glass material. On the right is a custom material with 100% transparency, fresnel reflectivity turned on, and an hdr environment texture. Both have the same base color and both have an IOR of 1.8. The intensity of the reflection can be adjusted with the strength % and the multiplier, and the hdr’s saturation can be adjusted to affect how much the color of the hdr affects the color of the gem.
For the diamond I changed the color to white and pumped the IOR up to 2.2 (the real IOR is higher, but I liked how it looked better at 2.2). I also reduced the strength of the env texture to 10%. But you can fiddle forever on these.
I use the peppermint factory hdr from polyhaven (free with a CC0 license).