Real Bible Translation

And the Woman, she was being a Pagan of Exalted Palm Country ("Syro-Phoenician") by the Race,²¹ and she was begging himself so that he should cast out the Demonic Power from out of the Daughter of herself.

Julia Smith Literal 1876 Translation:
And the woman was a Grecian, a Syrophenician by birth; and she entreated him that he would cast the demon out of her daughter.

LITV Translation:
And the woman was a Greek, a Syro-phenocian by race. And she asked Him, that He would cast out the demon from her daughter.

ESV Translation:
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Footnotes

The term Ἑλληνίς (feminine of Ἑλλήνιος) primarily denotes a Greek woman in classical usage, as evidenced in sources such as Euripides (El. 1076) and Menander (Men. 79). However, in later, particularly Christian-era texts, Ἑλληνίς also acquires the meaning of a pagan woman, as seen in Julian’s Epistles (Jul. Ep. 112). This reflects the polemical Christian redefinition of “Hellene” from an ethnic or cultural identity to a religious category signifying non-Christian or pagan status. Thus, the term’s semantic range expands from denoting Greek ethnicity to indicating pagan religious affiliation in post-classical usage.

(cf. LSJ)

Σύρος (Syros), i.e. ארם (Aram) (cf. Strong's #758 ארם)

The Greek term Σύρος (Syros), used to describe the region of Syria, is believed to be derived from or connected to the name Aram. The ancient Greeks used Σύρος or Συρία to refer to a region that included the lands of the Arameans (ארם), whose name comes from the root ארם (a-r-m), meaning "elevated/exalted" or "high."

Over time, Σύρος (Syros) evolved as the Greek designation for this broader area, encompassing not just the Aramean territories but also surrounding regions, reflecting the Greeks' adaptation of the local names and their own geographical understanding.

Φοινίκη Phoiniké - palm country (cf. Strong's #5403)

Phoiniké Phoiniké -