So, she stated that she wants me "out by mid-August". I enter my new place on August 1st. Since she initiated this, I don't understand why I am subject to moving out and paying rent at her convenience.
Imagine the scenario where I have lived in the house for less than one year, and she has given me 30 days notice. That would mean that I am forced to stay until the end of the 30 days, which give me, as the tenant, no flexibility whatsoever.
Additionally, the reference you stated only requires me to be OUT BY the end of 60 days.