Clearly her "mistake" was writing her own name first. Although I can see the logic in this having much less impact on Elsa's reputation than doing it the other way around.
I was about to point out, there's an interesting discrepancy between the manga and novel. In the novel, she writes "Elsa ❤ Yvonne", while here she writes "Yvonne loves Elsa". The former allows some ambiguity as to who wrote it, especially if the translator also translated the ordering of the English version. In particular, it allows the other students to assume that Elsa is the one going overboard with a weird crush. This ordering, however, makes it clear that Yvonne wrote it.
Interestingly, the novel ordering also makes it clear why Yvonne would want to avoid dragging someone else in, since putting their name first invariably opens them up to ridicule. Conversely, if Elsa's name were first, it would be fairly easy to keep the bullying targeted at her by making it look like she's chasing after someone who's "clearly" out of her league. Somewhat ironically, that probably would have worked with Yvonne, given her status at the start of the story, so she actively chose the least damaging combination, for Elsa, at her own expense.
Gee, I wonder why Elsa thinks Yvonne really loves her. ...
On a completely unrelated topic, I also have a new (highly unlikely) theory for why the story isn't going the way Yvonne remembers:
What if the original version was a Yuri light novel, not a hetero gacha game. The reason the version Yvonne is familiar with doesn't include that route is because the publisher pushed the author into making it a more profitable gacha game, and the original love interest got exiled to the hinterlands as a result. The System, then, is trying to enforce the Publisher's vision, not the original story.
Then again, Elsa does say "Aren't you in this situation because of me?", in both version, so it might be that they're both from the 'real' world and Elsa recognized the villain acting completely out of character and put 2 and 2 together.
