It seems like there is a bug when saving a map from ArcPro to AGOL using the Web Map tool through the Share Tab.
I created a new GDB, with a single feature class. Added five fields to the feature class. And added a single feature to the data.
I configured the pop-up to the following. Reversing the order of the fields.
Shared the map through the Web Map tool in the Share tab. Made no changes to the Web Map in AGOL. Then opened the Web Map in Field Maps. Started adding a feature and the field order reverted back to the order from the attribute table, ignoring the order set in the pop-up. However, the Pop-Up for added features in Field Maps is displaying in the correct order per the Pop-Up configuration.
Now, if I open the Web Map back in ArcPro through the Overview page of the Web Map,
Then use the Save Web Map tool in the Share Tab.
Then open the map in Field Maps, the fields are re-ordered to the correct order based on the pop-up.
I checked to see if clicking on Field Maps Designer would revert the order back to the Attribute Table order of One, Two, Three, Four, Five but it does not appear to be doing so. So, the bug appears to be at the initial sharing of the Map to AGOL.