If you ignore the trigger, you give up your readied action.
It means that you can take the readied action as a reaction after the trigger occurs, or not. If you ignore the trigger, you then forgo your readied action. You can take the Ready action again on your next turn. The language seems weird, but its intent is to specify that when the trigger occurs, you don't have to take your readied action. If circumstances have changed, you can choose to not take your readied action after all.
Offering the opportunity to "ignore the trigger" is all the rules have to say on the matter. Any disagreement about what that means will be up to the DM to adjudicate. In the example you provided, it would really depend on the trigger explained by the player, and the adjudication of what that means by the DM. If the player said, for example, "I ready a fire bolt to blast the first enemy that comes through the door," then I wouldn't allow the player to "hold" their readied action if the first enemy isn't one that they want to attack.
The rules say you can take your reaction or ignore the trigger. By offering no additional information, this strongly implies that you can't take another trigger, or change your trigger if you choose to ignore the first one. I think the rules would say so, if that were the case. The player is given two options when the trigger occurs, and that's it.