It's an interesting question, according to Marcus Langdon, a well-known author with deep interest in the history of the Straits Settlements. In an old account by the Penang Free School Historical Society which had interviewed Langdon quite some time ago (not later than 2008 or 2009, I believe), he had this to say about the school:
Q. When was the Free School founded and when was it first called the Free School?
Q. The name Penang?
A. The island was known as Pinang well before Light stepped ashore; Prince of Wales Island was purely the name given it by Light in August 1786 and it was henceforth officially known by that name, though in every day use it was still Pinang or Penang. All official records therefore refer to it as Prince of Wales Island.
Q. When was the school founded?
A. The first intake of students was, as stated above, on 21 October 1816, but one could argue that it was founded with the formation of the committee at the Council meeting of 6 January 1816. The early history is indeed not well known and is fascinating!