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I'm not worried about having to remove stuff manually if I change the cell style from my FigureCaption back to something else. That is a scenario that I would consider rare. So if that is a major concern, I'd say ignore it..

So this should work for you, add those (or analogous) styles to the stylesheet:

Cell[StyleData["Section"],
 CellDynamicExpression :> (
    SelectionMove[EvaluationCell[], Before, CellContents];
    NotebookWrite[
      EvaluationNotebook[], 
      TemplateBox[{CounterBox["Section"]},"TemplatedSection"], 
      After
    ];
    SetOptions[EvaluationCell[], CellDynamicExpression -> None]     
 ),
 MenuSortingValue->10000]

That part inserts what you want and disables this feature for given cell (because I don't trust CellDynamicExpression 126655).

It would be nice to not have to recreate all cells when the "header" is changed so the code above inserts only a TemplateBox for TemplateSelection which we can define later in the stylesheet and modify separately:

Cell[
   StyleData["TemplatedSection"],
   TemplateBoxOptions -> {
      DisplayFunction :> (
         TemplateBox[
           {#1,". section note",": "},"RowDefault",Editable->False
         ]&
      )
   },
   MenuSortingValue->10000 
]

I've noticed that sometime when the TemplateBox is edited cells are not updated properly/counter number is broken, but as soon as once creates one section and deletes it, everything is fixed. So that is only a minor inconvenience.