Hi @AdriannaBarton,
Thanks for pointing out the oversight in the documentation, we will have that fixed for the next update. The calculationMode actually applies to the first calculation, as well as subsequent recalculations. Therefore if you use manual, no calculation will be run the first time, until the button is pressed. The explanation of this was correct in the blog post, just not in the online documentation:
What's new in ArcGIS Survey123 (February 2022) - Esri Community
As for the issue you are seeing with your sum() calculation in the repeat, this is expected. It actually was not working correctly before, and values were not being updated when they should have been in 3.13 or earlier, a bug we fixed.
If you use a reference in a calculation to a field inside the same repeat, for a value that comes from the repeat or other records in that same repeat, the sum() will update after new values are added to other records. The best way to do what you are wanting, it to use position(..) and/or indexed-repeat() functions, to ensure the value is only updated once and for that current repeat record you are on. You can also use the once() function in combination to ensure it only runs once also. See below documentation:
https://doc.arcgis.com/en/survey123/desktop/create-surveys/xlsformrepeats.htm#ESRI_SECTION1_9F07ECB9...
If you search the forums there are many users also doing something similar to what you are after, and have successfully used position(..), indexed-repeat() and once() to achieve this across repeats.
Regards,
Phil.