At the moment there is only one way of generating summary from search page:
1) Click on "..."
2) Click on "Summarize Page"
It will create the summary inline (e.g. under the selected search result, without opening new tab) with default settings (Summary Type: Key Moments, Output Language: Default)
I propose to add a new item in the drop-down menu: "Open Page in Summarizer". It should work similar to "Open Page in Web Archive" - so it takes URL and opens kagi.com/summarizer with URL already filled in.

This will help users like me, who don't like default Summarizer settings, to be able to be already on Summarizer page where we can adjust "Summary Type" and "Output Language" as we want.
Right now to do the same I need:
1) Right Click on a Search Result
2) Click "Copy Link"
3) Open New Tab
4) type "kagi.com/summarizer"
5) paste URL in Summarizer
6) Change settings for Summarizer