Hi,
Thanks for the video, that actually helped a lot.
So in the video you are looking at an event called search_term. However that event is not coming from Ajax Search Pro. As you scrolled through the options on the back-end, the event names were the default: focus, search_start, search_end, magnifier etc..
The event search_term seems to be a core GA4 analytics event parameter. I suspect it is triggered when you view the search results page, when the enhanced measurments are active.
My guess is, that you should be able to test that by visiting the search results page with a couple of different phrases, they should start showing up under the search_terms event.