Is what the questioner did, by informing someone that she had ended her relationship with her friend to defend her reputation, considered prohibited backbiting, and is it permissible for her to defend her reputation by mentioning what happened to her out of necessity?

It is permissible for one who has been wronged to defend himself and clarify the truth, and this is not considered prohibited backbiting, due to the Almighty's saying: "Allah does not like the public mention of evil except by one who has been wronged. And ever is Allah Hearing and Knowing." (An-Nisa 4:148). Moreover, shunning and boycotting an oppressor is permissible if the intention is to reform him and deter him from his actions.