The GitHub - discourse/discourse-search-banner · GitHub is now built upon the glimmer search menu which is why I made the changes there instead of the widget search menu. Additionally, we will be rolling the glimmer search menu out as the new default for all discourse instances in the next month so to take advantage of the changes I have linked (and get one step ahead of the crowd) I would recommend enabling the glimmer search menu on your site. This will solve any issues you are having in the search menu located in the navigation bar.
You can do so by adding a group to the experimental search menu groups site setting. We currently have this enabled for “everyone” on meta.
The code snippets you linked are directly tied to the widget search menu. The same plugin outlets are available on the glimmer search menu, and they are correctly positioned. Let me know if that helps 