Let me try to sort it out for you.
Three Aryan or Indo-European tribes migrated from Central Asia into the area we now know as Iran in the 2nd or 3rd millennium BCE: the Medes, the Parthians, and the Fars. They settled in different parts of the Iranian plateau but spoke very similar Indo-European languages. Farsi was the language of the Fars whose land was known as Parsa which morphed into Persia. From about 500 BCE or so, the state-authorized religion was Zoroastrianism.
Remember Indo-Europeans are not Arab. Arabs invaded in the 7th century and proceeded to convert the Zoroastrians to the relatively new religion of Islam.
The Indo-Europeans called the country Iran, after the word Aryan or “country of the nobles.” But the rest of the world called the area Persia even though the Persians were only one part of the original ethnic mix. Reza Shah in 1935 let the world know that the country's name was Iran, to strengthen the concept of Iran's being one nation of different ethnic origins.
These days it is permissible to use the terms Iranian and Persian more or less interchangeably. I use the word Persia when I am referring to the culture of the country and Iran when I’m referring to the country, its government and policies.