In PDF, highlight is an annotation like strikethrough or underline; there is no difference.
The option you show in the screenshot concerns the applying of the highlight, strikethrough or underline tool. When you apply it, the selected text gets copied into the popup, and that's it.
You should, however be able to sort the comments list by type, and then you would have the highlights together.
If that is not an option, and it would be acceptable to have the list in a second document, then you can use some JavaScript to create a list of the annotations, filter out the highlight annotations, transfer that list to the second document where you can display the annotations in some form fields (including the dynamic link to the original).
I have developed such a solution for a slightly different application, but it would work in this scenario as well; feel free to contact me in private for more information and pricing.