If I understand you right I have the answer to your need, and it is simple. Below you see the screen shot where I show different formats created by my custom-made StyleSheet.

The first line is just the use of the inline formula , where the upper and lower limits are written on the side of the symbol, as you wish.

The second line is the cell DisplayFormulaNumbered. There I placed two sums one with the limits on side and one - above and below:

The code for the Inline formula is as follows:

Cell[StyleData["InlineFormula"], ScriptSizeMultipliers->{0.7, 0.7142857, 1.}, ScriptLevel->1, FontFamily->"TimesNewRoman", FontSize->12, FontWeight->"Plain", FontSlant->"Italic", FontVariations->{"StrikeThrough"->False, "Underline"->False}, UnderoverscriptBoxOptions->{LimitsPositioning->True}, FractionBoxOptions->{BaseStyle->{ScriptSizeMultipliers -> {1., 0.7, 0.7142857, 1.}}}] 

where I inserted your definitions, but kept few of my own. I also recommend to add the following cell into the StyleSheet:

Cell[StyleData["InlineCellEditing"], FontFamily->"Constantia", FontColor->RGBColor[0.502, 0., 0.502], Background->RGBColor[0.896, 1, 0.951]] 

This is not necessary, but helps during editing the inline cell.

Now to operate, you call the inline cell just by pressing Ctrl+9. This can be done from the text cell as well as from the DisplayFormulaNumbered cell. Then the slot of the InlineFormula opens. If you included the definition of the InlineCellEditing, this slot will have a different color, which helps during editing. Now you simply type the formula in the slot. For example, for the sun under script you simply type Ctrl+_ and so on. It will appear in the right place. If you need to insert the under/overscripts below and above the symbol, you just use the palette template: Menu/Palettes/Other/BasicMathInput.

I hope it helps.

If I understand you right I have the answer to your need, and it is simple. Below you see the screen shot where I show different formats created by my custom-made StyleSheet.

The first line is just the use of the inline formula inside a text cell, where the upper and lower limits are written on the side of the symbol, as you wish.

The second line is the cell DisplayFormulaNumbered. There I placed two sums one with the limits on side and one - above and below just for the purpose of the demonstration of such a possibility. Here the sum with the limits on side is done within the InlineFormula placed inside the DisplayFormulaNumbered cell. The second sum (with the indices above and below) is in contrast typed in the DisplayFormulaNumbered cell.

The code for the Inline formula is as follows:

Cell[StyleData["InlineFormula"], ScriptSizeMultipliers->{0.7, 0.7142857, 1.}, ScriptLevel->1, FontFamily->"TimesNewRoman", FontSize->12, FontWeight->"Plain", FontSlant->"Italic", FontVariations->{"StrikeThrough"->False, "Underline"->False}, UnderoverscriptBoxOptions->{LimitsPositioning->True}, FractionBoxOptions->{BaseStyle->{ScriptSizeMultipliers -> {1., 0.7, 0.7142857, 1.}}}] 

where I inserted your definitions, but kept few of my own. I also recommend to add the following cell into the StyleSheet:

Cell[StyleData["InlineCellEditing"], FontFamily->"Constantia", FontColor->RGBColor[0.502, 0., 0.502], Background->RGBColor[0.896, 1, 0.951]] 

This is not necessary, but helps during editing the inline cell.

Now to operate:

you call the inline cell just by pressing Ctrl+9. This can be done from the text cell as well as from the DisplayFormulaNumbered cell. Then the slot of the InlineFormula opens. If you included the definition of the InlineCellEditing (given above), this slot will have a different color and font, which helps during editing. Now you can simply type the formula in the slot. For example, for the sum underscript you simply type Ctrl+_ and so on. It will appear in the right place. 

If in the DisplayFormulaNumbered cell you need to insert the under/overscripts below and above the symbol, you do it outside the InlineFormula cell. Close it, if it has been open. Just use the palette template: Menu/Palettes/Other/BasicMathInput.

I hope it helps.