I plagiarized in my Bachelor's thesis, and after a year,went back and redid it unofficially. Should I push for some official action? My supervisor does not want to push this further and take official action since this was a lone incident and I had health issues at that point. However, since I am not getting closure, I do not feel like I have a go-ahead to re-apply to universities. I feel that future advisers/people I write to for grad school should be aware of this but do not know how to bring it up, since there's no official record of this. By plagiarized, I do mean that there's no question/argument about it, there's some straight up copy-paste. I have reworked it now, but feel like I have put an end to further education for myself. I also turned down some grad school admits (this + monetary issues). Any advice/clarity would be appreciated.
No, don't push, and don't tell it to future advisors etc.

You shouldn't feel this obligation to offer up the truth, especially about bad things, to people who don't even want or need to hear it.

Lying and openly telling all of your problems are totally different (especially when not even asked about them, by people who don't even want to hear about them, who have the power to make decisions that will be harmful to you).

Your supervisor thinks you are a good person, and they don't want to make unnecessary problems for you. You've even shown your character by going back and re-doing it. If your supervisor thought you were lying or that you would keep doing it in the future or that other people would be harmed by you elsewhere, they would want to make official problems for you.

So what I mean about lying/deception vs openly telling everyone all of your problems: there are people who are always only thinking about getting to the next step, and will lie to their classmates/coworkers, or steal from them etc. If some interviewer asks you "did you ever plagiarize anything?" then you can tell them (but if you believe they shouldn't even ask that question, or that the answer is irrelevant, then feel free to lie). But if you apply for a PHD or something and they ask about it, then you should probably tell them, and tell them the whole story.

And I don't mean you have to let them ask exactly the right question. If they ask something that seems like it's related, and that they would want to know this, feel free to tell it. I just mean, if they ask some general question about your history, you shouldn't tell them every bad little thing. Would you tell some random people at the bus stop too? "Hey man do you go here?" "Yeah I go here, but I plagiarized this thing like 2 years ago, I thought you should know". You know what I mean?

Just forget about it. Even the people in charge of dealing with it don't want to keep going with it. Their job is to make official problems for people who are doing it as part of a pattern and who don't want to correct their actions, to prevent future harm to other people.