I think you are going to have to say what your laptop, operating system and browser are in the hope that someone with a similar set up can investigate.
On my MacBook, with Safari, I have just reproduced the behaviour that I described above for my iPad—at an intermediate window size I can force a view that looks like the view that you posted above, with no search modification options. Are you changing window sizes at all? If so, try increasing the window size to see if the FindMyPast sidebar reappears.
Also I noticed that in my set-up, where Safari has its own sidebar (to display tab groups, bookmarks etc), toggling that to visible can modify the FindMyPast window to transition from full FindMyPast side bar visible to the state with no options.