The key word in question is מוֹלֶדֶת (moledeth Strong's H4138), which is defined as "kindred, birth, offspring". This could mean that Abraham is referring to either where his parents were from or where he was born.
To my knowledge, there is only one "Ur". In the topic of "Ur" in the Insight on the Scriptures, we see its identified location:
2. “Ur of the Chaldeans,” the city in Mesopotamia where Abram’s (Abraham’s) brother Haran (and likely Abraham himself) was born. (Ge 11:28; Ac 7:2, 4) Jehovah appeared to Abraham and directed him to leave Ur. The Bible, crediting Terah with the move because he was the family head, says that Terah took his son Abraham, his daughter-in-law Sarah, and his grandson Lot, moving from Ur to Haran.—Ge 11:31; 12:1; Ne 9:7.
Usually Ur is identified with Muqaiyir, which is W of the present bed of the Euphrates and some 240 km (150 mi) SE of Babylon. Ruins there cover an area that is about 910 by 730 m (3,000 by 2,400 ft). Once a center of worship of the moon-god Nanna (or Sin), the site’s most prominent feature is still a temple tower, or ziggurat, some 61 m long, 46 m wide, and 21 m high (200 by 150 by 70 ft).—PICTURE, Vol. 2, p. 322.
Note that the article mentions Ur as possibly being the city of Abraham's birth. "Haran" is both the name of a person (brother of Abraham) and a place (city of northern Mesopotamia).
Abram (later Abraham) was probably born and raised in Ur. Afterward, Jehovah God tells Abram to leave Ur and move to a different land. (Genesis 12:1) On his way to the land of Canaan, Abram and his family (including his father, Terah) live in a city that is called Haran. (Genesis 11:31, 32)
In Genesis 24, we find Abraham, now well advanced in age, giving instructions to his servant to go find a wife for his son Isaac. Abraham directs his servant to where his current relatives are living. In verse 7, Abraham tells his servant "you will certainly take a wife for my son from there" meaning Haran.
So, while "land of my nativity" may be a reference to Abraham's birthplace, the instructions he gives to his servant are to the current city of his relatives, Haran.
[All scripture quotations from the New World Translation of the Holy Scriptures (Study Edition)]