I think it's clear that Elsa doesn't intentionally unleash an unseasonal winter on her kingdom. Once she finds out that this is happened, she is horrified, tries vainly to think of a way to stop it, and finally attempts to get away from Arendelle altogether hoping the winter will go with her.
However her song on the mountain reveals a certain amount of frustration and possibly resentment. She expects hostility and rejection from everyone, but it doesn't bother her anyway. She will hold their reactions in contempt.
Out of universe this is because the song was intended to be a villain song, explaining how and why she wasn't going to be bound anymore by petty restrictions like not retaliating with magical snow-blasts to people calling her names, etc. In universe we can say she had reasons to not feel positive toward the society she was running away from, and her song reflects those feelings at that point in time, but not necessarily anything that could be called deliberate ill will.
So: Is it likely that the July winter storm was connected with Elsa's possible negative feelings toward Arendelle, which she may not have been fully aware of at the time? Or was it entirely random and not directed at any specific place or people, whether consciously or unconsciously?