That activated ability can be activated only once, so it is impossible to repeat the choice and "that hasn't been chosen by you this game" becomes functionally irrelevant text.

If you want something that e. g. tracks the modes chosen after leaving and returning to the battlefield of a commander, then you need to present a different wording that explicitly explains that you track the modes.

I wonder whether some new tech like anchor words would be worth introducing for this, but - as is - the rules do not accommodate an easy wording for such an effect.

Maybe adding "for an activated ability of a permanent named The Cinder King"? Clearly name-changing effects and ability sharing effects exists that can meddle with this, but they are rare.