Was Amasa’s father Ithra the Israelite or Jether the Ishmaelite?

The name is the same in both. Ithra (Hebrew Yitra) and Jether (Yeter) are two forms of one name; a similar pair is used of Moses’ father-in-law, who is Yeter and then Yitro in a single verse (Exodus 4:18, where the KJV has Jethro both times). Both passages also name the same mother, Abigail, the sister of Zeruiah (and, by 1 Chronicles 2:16, of David), so they plainly mean the same man.

The nation needs more explaining, and two answers are given. Keil and Delitzsch point out that there was no reason for Samuel to say that Amasa’s father was an Israelite, since nearly everyone in the story was; it is worth saying only if he was not. They conclude that ‘Israelite’ is a copyist’s slip for ‘Ishmaelite’, two Hebrew words with several letters in common (see textual), and that Chronicles preserves the original. John Gill, with the Jewish commentators Rashi (Jarchi) and Kimchi, suggested instead that both words are true: Jether was an Ishmaelite by birth who had joined Israel, or an Israelite who had lived among the Ishmaelites. David’s own servants included “Obil the Ishmaelite” (1 Chronicles 27:30), and a man could be named from the place he had lived, as “Obed–edom the Gittite” was (2 Samuel 6:10). Either way the two verses describe one man.