Philistia or P'lshet is the region, also means the foreigners.
Philistim or P'ilishtim means the Philistine people.
The difference between a P and an F in old Hebrew is a tiny dot, which could easily get lost making an F into a P. But the Septuagint, which predates the Masorete vowel pointings considerably, puts an F [or Ph in Greek] as the first letter, at least in the places I looked.
Another term used for Philistine is Gentile, as in Islands of the Gentiles, which means the same thing = the People, probably foreigners.
The Egyptian term is as you said 'plst', which probably means Sea People.
They did emanate from the Aegean Sea area and came by boat to some degree, but they advanced through Palestine on foot laying waste to city after city, their families following in oxcarts. Even chariots didn't stop them. Originally they were from the area around the Black Sea, although this is probably a guess since they weren't noticed by history until they started tearing up the Middle East. They had iron age technology, which gave them a distinct advantage in the Middle East and would have in Egypt, too except they were trapped on their ships and shot up by Egyptian archers.
English translations may be causing some of the linguistic problems, and I am probably causing the rest.