high_resolution_icon_url... But I wasn't able to find any information about any of those images in the documentation. Are any of them fixed size? And/or is there a place where I can find the intent for each of those images?
I can't really answer that question directly, but for reference, the dimensions of the images supplied by the API for Stack Overflow are as follows:
Image: Dimensions: ===================================== logo_url 250x61 icon_url 158x158 favicon_url 16x16 high_resolution_icon_url 316x316
I didn't have the patience to verify if the other sites used the same dimensions - but I would assume that they do.
There is nothing in the documentation (that I am aware of) indicating specific intents for these images. But here is my humble opinion on each of the images: