Hover mouse over the Ref. or Value and use hotkey M, then move the text to wherever you wish.
If you are using symbols from the Kicad library, maybe that symbol has had the text moved since your first connector was placed. Maybe you moved the text and you have since forgotten.
The text is placed as it occurs in the Kicad library. If you want the text to always appear in a certain position that is different to the Kicad library position, you must first copy the symbol into a personal library, then alter the position of that text.
That looks like a generic connector symbol, so 2.0 mm JST-PH isn’t the original value text, it would be Conn_01x06 or something like. It could be that by design if you are not reverting to the original text, it doesn’t revert to the original position either. Dunno, I never worry about the original positions since it’s almost always unsuitable and I have already changed them.
So it seems that when I run “Update Symbols from Library” then J2 is actually taking the positions from the library but for J9 the texts are neatly arranged at the right side. Where does the difference come from?