In that case I think I know what the problem might be. Do you happen to use any cache plugins? Because someone about a few weeks ago had a similar issue, and it was related to how the page was cached, and since then I was able to find a solution as well.
It's related to a fall-back method that the plugin uses whenever someone with a very very old browser (namely Internet explored 8 and below) visits the site. Then the search shortcode is replaced with the default search engine automatically. Now, some cache solutions might not create different versions of the cache for old browsers, thus showing the fall-back search for all browsers.
The solution I came up with was simply an option on the Compatibility Settings submenu: https://i.imgur.com/zX2d5uu.png
After turning that option OFF, make sure to clear all your website cache. After that, the ajax search pro search bar should start appearing everywhere as expected. If not, then it's still cached somewhere – page cache, CDN cache, cloudflare etc.. – so make sure to clear them all, if you use them.
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