Hey there k-705,
Upon researching your question, we did not find any information about 'HDR badges' for photos, but we did find a link that goes over how HDR works, along with what options you should expect to see for HDR as it'll depend on your model(iPhone 12): Adjust HDR camera settings on iPhone
By default, HDR is enabled on your iPhone, and is used when it’s most effective. For your model, the link also goes over the steps to take if you wish to disable automatic HDR for photos. Also for your model, by default, the HDR version of a photo is saved in Photos. Only in iPhone X and earlier you have the option to keep a non-HDR version of a photo.
In case you'd like to provide feedback to Apple about this for consideration, you can submit it here: Product Feedback
Cheers!