You're correct.
In most contexts the term 'firstborn' means the first son in birth order.
I some contexts the term 'firstborn' is a title of rank, inheritance, preeminence and superiority.
Manasseh was the first son born of Joseph, yet GOD calls Ephraim His firstborn.
Gen_41:51 And Joseph called the name of the firstborn Manasseh: For God, said he, hath made me forget all my toil, and all my father's house.
Gen_48:14 And Israel stretched out his right hand, and laid it upon Ephraim's head, who was the younger, and his left hand upon Manasseh's head, guiding his hands wittingly; for Manasseh was the firstborn.
Gen_48:18 And Joseph said unto his father, Not so, my father: for this is the firstborn; put thy right hand upon his head.
Jos_17:1 There was also a lot for the tribe of Manasseh; for he was the firstborn of Joseph;
Jer_31:9 They shall come with weeping, and with supplications will I lead them: I will cause them to walk by the rivers of waters in a straight way, wherein they shall not stumble: for I am a father to Israel, and Ephraim is my firstborn.
1Ch_5:1 Now the sons of Reuben the firstborn of Israel, (for he was the firstborn; but, forasmuch as he defiled his father's bed, his birthright was given unto the sons of Joseph the son of Israel: and the genealogy is not to be reckoned after the birthright.
Ishmael was the first son in birth order but the rights and inheritance belonging to the firstborn went to Isaac.
Esau was the first son in birth order but the rights, blessings and inheritance belonging to the firstborn went to Jacob/Israel.
Exo_4:22 And thou shalt say unto Pharaoh, Thus saith the LORD, Israel is my son, even my firstborn:
David was the youngest, far down the line from the first in birth order but GOD says that He would make him his firstborn, higher than the kings of the earth.
Psa 89:27 Also I will make him my firstborn, higher than the kings of the earth.
This Psalm not only pertained to David but is Messianic prophecy which pertains to the Lord Jesus and is repeated of Him in Revelation.
Rev 1:5 And from Jesus Christ, who is the faithful witness, and the first begotten of[ek/out of] the dead, and the prince[archon/chief/head] of the kings of the earth.-KJV
Rev 1:5 and from Jesus Christ, the faithful witness, the first-born out of the dead, and the ruler of the kings of the earth- YLT
Then, we have the other three which pertain to Christ's preeminence over the creation and over the new life from out of the dead.
These do not pertain to His origin or some fanciful imagined beginning of existence.
Rom_8:29 For whom he did foreknow, he also did predestinate to be conformed to the image of his Son, that he might be the firstborn among many brethren.
Col_1:15 Who is the image of the invisible God, the firstborn of every creature:
Col_1:18 And he is the head of the body, the church: who is the beginning, the firstborn from the dead; that in all things he might have the preeminence.