"branch" paragraph style, send changes through document

I'm not quite sure I follow your steps, but —

• A style named Title will affect all paragraphs it's applied to, as long as there are no overrides, either spot formatting or Character Styles. (This gets a little tricky if you use sub-styles like GREP or Line formatting within a style, but I suspect you're not.) Put the text cursor in each example of Title; the Title paragraph style should highlight in the list, with no Character Style also highlilghted and (most important) no + mark on the end of the Para Style name, indicating an override has been applied (local color, size, spacing, etc.)
• Select the one instance of Title you want to tweak. Duplicate the Title style with the + icon at the bottom of the Para Styles list. (Yes, there are too many '+' signs in this app! 🙂 ). Select the new style, name it Title_Paper or equivalent, and make whatever change you want to it. Save the style and make sure it's applied to that instance with no overrides etc.

Title_Paper should now follow Title in all respects (changes, etc.) except for those further changed in Title_Paper.

Be sure you are changing the actual paragraph style, not one applied to a text frame. They're not always quite the same thing.

Check back if that doesn't work right for you.