The beginning of Shemot 14:5 says, "It was told to the King of Egypt...". Why use this term instead of the more common term "Pharoah"?
I don't have an authoritative answer but one comment that the Ohr Hachayim makes made me think. He cites an opinion (and, admittedly, the abbreviation for the source is unfamiliar to me) that Par'oh wrote a get shichrur for the nation, freeing them. If that is taken as a fact, then the people who informed Par'oh would have to have grounds to appeal to him -- but Par'oh can't go back on the contract he signed (yes, I'm borrowing liberally from the kingly notion of ein lehashiv from the megillah).
So the text indicates that there is this loophole the king tries to employ by being thought of as another "personality", the 'king of Egypt' as opposed to the "Par'oh" who signed the document. But the later use of Par'oh in the pasuk, and the use of the combined terms in pasuk 8 allow the text to destroy that loophole: the two personalities are one and the same person and he should be bound by his signature. Semantic games can't be employed to give him deniability.
Of course, I could be completely misunderstanding the Ohr Hachayim, and am just making this up to explain what could be a simple stylistic textual choice. Take it with as much salt as you need.
Without any sources, I think Pharoh is how you perceived Pharoh subjectively when you are in Mitzrayim. He is the Pharoh, he defines society and there is nothing else.
Once you have left Mitzrayim then you know objectively that there is a country called Mitzrayim which has a king.
This was the essence of the entire geulah, that the avadim should know there is more in the world than this Pharoh who was eating into their soul.
Perhaps the text is suggesting a direct action on the king and not on his court and servants. Historically and etymologically the word "pharaoh" actually means "Great House", that is, the palace of king, not the king himself. In this sense "king of Egypt" would be more especific (although without naming) depending on the usage in its time.
According to Joel Forman in The use of the Term Pharaoh in the Bible (Jewish Bible Quarterly, Vol. 43:1 (169) January – March 2015:
The kings of Egypt were not originally called pharaohs by the ancient Egyptians. The term "pharaoh" for the king of Egypt developed over time, and was also used by the Hebrews and Greeks to describe the Egyptian ruler. In Ancient Egypt, the term "pharaoh" was not originally a royal title. Translated literally, the earliest meaning of the Egyptian word per-o was "great house", that is the palace or residence of the king and his administration. The term "pharaoh" referred to the ordinances and commands the king issued in his administration, but not to the person of the king himself. In New Kingdom times (sixteenth century BCE), it began to designate the king himself, rather like our use of "The White House" to refer to the American president or "The Crown" to refer to the British monarch.