Whatever term you use, as you perceive it, there is the possibility of ambiguous interpretation (to abort something, to put an end to it before it is finished, therefore to cancel);
terminer: Faire cesser quelque chose; mettre un terme, mettre fin à quelque chose. Synon. clore, finir.
There is however a verb that answers specifically to your requirement, although it might not be congenial to you (it doesn’t appear on screens as far as I know); it is the verb “achever”.
It is unclear what is saved when the user click on “finish up”, when does the button appear, and what other choices are available.
In any case, what can be done is to use two buttons, one labelled Annuler, Quitter or Abandonner which would cancel whatever is expected to be completed, and display a pop-up window asking for confirmation if clicked, and a second button labelled Suivant or Continuer that would display a different screen when completed.
“J’ai terminé mon examen” means you complete it. Like You start from the start and end by the end.
“J’ai finis mon examen” means you end the examen but you don’t have clue if you complete it or not, maybe you write nothing on your exam. Like you just end by the end.
So in your case I will feel more confortable with a terminer. Means that the job will be done even if I click on the button.