Did Eliezer fetch a wife for Isaac?

1 Now Abraham was old [H2204], well advanced in years. And the LORD had blessed Abraham in all things. 2 And Abraham said to his servant, the oldest [H2205] of his household, who had charge of all that he had, “Put your hand under my thigh, 3 that I may make you swear by the LORD, the God of heaven and God of the earth, that you will not take a wife for my son from the daughters of the Canaanites, among whom I dwell, 4 but will go to my country and to my kindred, and take a wife for my son Isaac.”

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According to most interpretations, the unnamed "servant, the elder of [Abraham's] house, that ruled over all that he had," (Genesis 24:2) who obtained Rebeccah as a bride for Isaac, was the same Eliezer. This name is first found in the Bible in Genesis 15:2 when Abraham asks God about his promises of a Son while being childless, naming Eliezer being the current heir of his house.

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2 But Abram said, “Sovereign LORD, what can you give me since I remain childless and the one who will inherit my estate is Eliezer of Damascus?”

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