A multilingual stele from al-Kabrit (Saudi Arabia) dating from the reign of Darius I (550–486 BC) provides some of the modern confusion here (but it would not explain Strong as it was discovered long after Strong). The stele refers to Put (sometimes spelled Phut) in three languages. Both Old Persian and Neo-Babylonian connect Put with a region of Persian. However, the equivalence line of text in Egyptian places Put as part of Libya.
You are correct in that most of the references to Put indicate they were from the region we know as Libya. The Septuagint of Ezekiel uses Libues whenever the Hebrew has Put (27:10, 30:5, 38:5).
27:10 Men of Persia, Lud, and Put were in your army, men of war.
They hung shield and helmet on you; they gave you your splendor. (NET)
30:5 Ethiopia, Put, Lud, all the foreigners, Libya, and the people of the covenant land will die by the sword along with them. (NET)
38:5 Persia, Ethiopia, and Put are with them, all of them with shields and helmets. (NET)
Possibly, Strong placed Put as part of Persia because two of these three references link it with Persia (27:10 and 38:5). Likewise, Lud
(27:10 and 30:5) refers to a settlement near the Tigres. However, 38:5 and 30:5 also link Put with Ethiopia, part of Africa. Obviously, all three cannot be linking them by geography. The places are linked for other reasons in these texts.
Josephus (Antiquities 1.6/2), Pliny the Elder (Natural History 5.1), and Ptolemy (Geography iv.1.3) all refer to Put as being west of Egypt. In addition, Ptolemy mentions a city named Putea in Libya (iv.3.39). Josephus writes that the region was once known as Put but the name has been changed to reflect the name of a son of Mezraim, Lybyos. Mezraim being the Hebrew name for Egypt.