Hi,
This looks like a cache related issue to me. The search bar on the second picture is actually not the plugin search – so the cache might be serving an outdated version of the site.
To make sure it is no a browser user agent mix-up, please turn off this option: https://i.imgur.com/Dsz45Tf.png
After that, clear all layers of cache and it should be fine.
I have only seen this issue once before, and then it was related to that option. Some google robot visits were cached as if real page loads and the search was not rendering to them, and that cache was served back to the users then.