In the first three films (ignoring KOTCS for this), Indy is a dynamic character. In TOD, the diamond is instrumental in portraying this.
In Raiders, he is obsessed with getting the prize - the idol in the introductory adventure, the Ark in the main story. Leaving Marion in the tent to continue his recovery of the Ark is the primary demonstration of this. And while he later pronounces his willingness to destroy the Ark to try and save Marion, he is shown to be weak - he can't follow through with this. In contrast to the later films, he's the most static in Raiders.
In Crusade, he's also obsessed with the prize. In the teaser(s) it's the Cross of Coronado - "It belongs in a museum!" It's shown that he inherits his obsession from his father, who spent his life obsessed with the Grail. While Indy's motivation for most of the film is simply rescuing his father, at the end he is shown overcoming his obsession (with his father's help) and letting go of the prize for the sake of family.
In both of films, the change in the character is done in a subtle way. In Temple of Doom it's very explicit. Fast forward to the village in India, and Indy's primary motivation is simply to get to Delhi until he learns about the Sankara stones - gigantic diamonds. While he shows sympathy for the plight of the villagers, he explicitly uses the phrase "fortune and glory" to describe his motivations in setting off to Pankot. When he takes the stones from the temple, his motivation is portrayed as greed - there's no indication that he would actually return a stone to the village! It's only when he discovers the enslaved children that he starts to think of helping others, and after his recovery and the plea from Willie to get out of there, promises "All of us" and goes back after the children. Finally, he returns the stone to the village, respectful of the villagers and dismisses the potential "fortune and glory" of the Sankara "stone".
Keeping that in mind, look back on the opening scene. He's indeed trading away an artifact for "fortune"; presumably the diamond is worth more than Nurhachi's remains.
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It should be pointed out that there is also a later bit of reconning. In The Lost Journal of Indiana Jones the diamond is claimed to be the Eye of the Peacock, a treasure Young Indy hunted for in 1919. If this connection is accepted, it puts a very different spin on Indy in Temple of Doom. In this case, Indy's interest in the Nurhachi trade is not simply for profit embodied in a gemstone, but for a notable artifact in its own right, one that he's been in search of for 16 years. While this is a nice connection to make, it someone devalues the character evolution portrayed in Temple so I'm on the fence about it. (As a fan of YIJC I find it provides a bit of closure for long suffering Remy, but it detracts from the strength of TOD's story.)