References comparing the two different definitions of joule expansion coefficient

Some pieces of literature derives Joule thompson coefficent using enthalpy $H$ and others derive it using internal energy $U$ … so which one is the right one? If they are both equivalent, then how do you prove their equivalence?

Edit: Ok, I realized there are actually two different JT coefficents called the JT coefficent. So, I seek an answer which discusses both.

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Sources which derive using internal energy:

Lecture -3 of mit ocw around 49:14

Page-63 of Atkin’s physical chemistry (internal edition)

Robert G. Mortimer – Physical Chemistry (page -63) for experimental explanation of this phenomena

Sources which derive using enthalpy:

Joule thompson coefficent in wiki

Charles Kittel, Herbert Kroemer – Thermal physics-W. H. Freeman [ page 362, chapter-12 cryogenics]

In chemistry libre text