using a calculated column with [Tab:TabName] would be more straight forward.

The custom XSL is not lost, the script is still inside the XSL script file. UDT stores all viewing parameter inside the tab settings, and this offers the most flexibilty to UDT.

However I recognize your problem, maybe we should add a button "save current settings as default" which would copy the tabsettings to modulesettings.This would help in all copy scenarios. This would also help if you enable "show on all pages".

Hint for developers: 
Settings are really cool. You can ask for a setting withs Settings("SettingName"). Settings are merged from tab settings and module settings, where tab settings overwrite module settings with the same name. We currently save all viewing settings inside the tab settings. This allows to present the same data in different ways on different pages. However, if UDT is getting reused on a new tab, or gets imported/exported, the tabsettings get lost and UDT initializes again with the default parameters inside code. However, as UDT always reads from settings instead directly from tab settings, a copy of the tab settings to the module settings would cause the initialisation with your saved settings, and as we always write  to tabsettings, can be overwritten again. 

The XSL based search is different as it is Case aware. Please try again