I’m not even sure what is considered right or wrong in this situation.
Your supervisor has done the wrong thing here. When you publish a paper jointly, the submitting author who is dealing with the journal has a responsibility to ensure that all authors consent to the publication and are okay with the final form of the published work. Exceptions to this should be raised with the journal editor and dealt with appropriately. That is a standard expectation in academic publishing and journals typically (always?) require the submitting author to confirm this as part of the submission and/or publication agreement.
The wrong done here is mitigated (but not entirely) by the fact that you were put on notice that they were proceeding without you and you apparently raised no objection to this. It is not great to bypass an author with an “It’s fine, we’ll handle it”, but if you didn't raise any objection to this then your supervisor may have thought that you were okay with this.
So, the wrong thing has happened here. It is not how academic submissions are supposed to be conducted and it is likely that your supervisor is in breach of their agreement with the journal, and possibly in breach of academic research rules of their institution. On the scale of academic wrongs, this is not the worst infraction ever and is probably only a mild breach, but it is poor practice and it has led to an outcome where one of the authors of the paper is unhappy with the published work, which is a great shame. It has also evidently caused some incorrect data/results to be published, which will require an erratum to the paper.
I honestly don’t understand why they took my original work, changed it, and published it in the way they wanted, as if it were theirs.
When supervisors do academic work with students, it is typically the case that the supervisor is experienced in academic publication and the student is not (as is probably the case here). Academic supervisors are typically very skilled in revising work to make it ready for publication and then dealing with submissions systems. Academics also have a lot of experience in judging what level of contribution warrants credit for a relevant section and/or co-authorship of the paper.
Your supervisor probably thought that taking on these tasks without requiring further input from you was a favour to you, and would give you a joint academic publication without you having to do further work. (They probably also did it to save the time/hassle of coordinating with multiple authors.) Since you are a masters student, they probably figured that having a joint publication would be a boon for you and would help to kickstart your publishing career.
I’m also considering applying for a PhD, and I need my supervisor as a referee, so I’m afraid of damaging the relationship. ... [F]or PhD applications, if I say that my thesis project was entirely mine and that I led the full research process, the paper lists “Conceptualization, Formal analysis, Writing – original draft” as shared among multiple authors.
Whilst there are possible avenues of complaint open to you (through the university or journal), this might be one of those cases where you are better off taking this on the chin. Your supervisor has not done the right thing here, but it is a low-level infraction, and you still have a joint publication to your credit (which will be valuable to you in your PhD application). It is unfortunate that you are not happy with the final product, but you might take some consolation in knowing that most experienced academic researchers look back on their thesis work and resulting publications as being of lower quality than what they do later.
For a PhD application, irrespective of this event, it would be wrong to say that your thesis work was entirely yours anyway, since supervision and review by a supervisor is typically considered to be a genuine academic contribution to the work. You are better off just saying that you performed research under the supervision of your supervisor and this manifested in a joint paper published in a journal. Having completed your MSc thesis successfully, plus having a published paper with your supervisor, puts you in a good position for your application.
Finally, if there are indeed serious errors in the published paper then you and your supervisor should work on an erratum to correct this, or retract the paper if necessary.