I found this on a site, hopefully answering that question we had in class.
The term “Filipino” is actually of Spanish origin and not in anyway rooted itself from English. Since the name of the Philippines was ‘Las Islas Filipinas’, Philippine-born spaniards were called ‘Filipinos’; meaning, they were born from these islands. During the colonial times, this term is only exclusive to the insulares/criollos and filipino mestizos; but in modern times, filipinos adapted this concept, which is anyone born from these islands is a Filipino, and the term ‘filipino’ made its way to the english language. So that’s why it’s not spelled as ‘Philipino’ because the term did not come from the renamed english name Philippines.
The tagalog term is ‘Pilipino’ because the tagalog alphabet has no ‘F’; and since it is phonemically related to the sound of ‘P’, it was then adapted in that way. Before, the term ‘Pilipino’ can both refer to the people and to the language they speak and it was only in 1987 when The National Language Institute changed the letter ‘P’ into ‘F’ to create distinction between the people and the language in terms of its usage; along with the addition of foreign-acquired letters (including ‘F’) in the modern Filipino alphabet. However, that is only the case when the language is in Filipino. In English, the term ‘Filipino’ is still referent to both the people and the language they speak. and here’s an example:
Ang pambansang wika ng mga PILIPINO ay FILIPINO.
The national language of FILIPINOS is FILIPINO