NO -- it doesn't apply in your case. You are not an F-1 student. You are an AOS applicant. As such, EAD is required for any sort of work, including internship. In fact, even F-1 students are strongly recommended to apply for EAD. Scroll down the page you linked to see this note -- "Given the difficulty and ambiguity of the DOL six-factor test, the safest rule for international students should be that, if a student is providing a beneficial service to an employer, whether paid or unpaid, the student should obtain proper authorization to work."