Context selector menu - use instant filtering and sticky search bar

[adamviktora]

Context selector menu examples should be updated in context selector and in the custom menus too. Right now, it has these problems:

1. does not use instant filtering, you have to click on the search button instead (even enter key does not work)
2. when scrolling down, search bar will not stick at place

Screen.Recording.2023-11-08.at.9.31.23.mov

Expected behavior
Similar to this With inline search filter example

After merging #9687, and fixing this issue, we would have both With inline search filter and Context selector menu examples in Custom menus page, both with the exact same behavior. So we should probably keep only one of these examples and delete the other one, it's questionable whether name it With inline search filter or Context selector menu

[wise-king-sullyman]

Thanks for taking the time to write this issue up, but it sounds like this is the intended behavior of the context selector and the leads don't think it needs to be changed.