Search Windows opened from the Project Browser and Windows Menu act differently

Steps to reproduce:

1. Create a "Universal 3D" Project
2. Open the Search Window from the Project Browser button
3. Observe that you are searching "in | all" and type "scene" in the search field
4. Observe how many search results are in the displayed in "All, Project, Menus..."
5. Open Saved Searches Panel (F3)
6. Save the search for User by clicking the save icon and observe how many search results are displayed to the right of the newly saved search
7. Click on the newly created search in the Hierarchy on the left to reload it and see the number change
8. Now open a new Search Window from Window > Search > New Window and try and perform steps 3-7

Actual results: Searches saved from the Search Window opened from the Project Browser and a new Search Window act differently. If this is as expected there are a few issues:

1. The Search Window opened from the Project Browser immediately after the Project is created still displays different columns for "in | all, in | Project, in | Files" (this shouldn't be the case if the Search Window is for some reason set to just search for in | Project by default) This issue disappears after closing and reopening again from Project Browser, but users are then unable to search "in | All" after opening this window from the Project Browser

2. Searches saved for User in the Search Window opened from the Project Browser display results if the User was searching in All, but change the result number to in Project after clicking on it

3. Selecting a search saved from a Search Window that is opened by going to Window > Search whilst in a Search Window opened from the Project Browser (in the repro video 30 results) "converts" the Window as if it was opened from the Project Browser and selecting searches saved in the Project Browser Search Window no longer shows In Project Results (in the repro video 2 results, will now show 30 results)

Expected results: If this difference between the type of Windows is expected (which doesn't seem right, as there is no indication for the user that these Search Windows are in any way different) there should be some kind of an indication to our users that this is happening, and these cases dealt with accordingly. If it is not expected, there should be no difference between how these Search Windows are opened and they should operate in the same way.

Reproducible with versions: 2022.3.29f1 (Project Browser Search only uses "in | Project" search, searches saved in both Search Windows show different results, 27/2), 6000.0.0b14, 6000.0.4f1

Not reproducible with versions: 2021.3.38f1, 2023.1.20f1, 2023.2.20f1, 6000.0.0b13

Tested on (OS): Windows 11, macOS Sonoma M1