The word specifically used by Philodemus is πειθαρχησομεν and translator of "On Frank Criticism" translates it as "obey (one in authority)" which is in line with the LSJ definition. However, the word is related directly to:
πείθω
to convince, persuade
to succeed through entreaty
(mediopassive, πείθομαι, and Epic future, πῐθήσω, with dative)
to obey, yield
to believe, trust
(second perfect active, πέποιθᾰ, with passive sense) to trust, rely on (with dative of person or thing)
(perfect passive, πέπεισμαι, post-Epic) to believe, trust (with dative)
Which, I believe, adds necessary nuance to that "obey" and gets closer to being convinced there is a reason to trust, believe, rely on the person or school or "authority" (αρχή) by which one chooses to live one's life.