The definition of reincarnation is the rebirth of a soul in a [new] body. Jack still looks the same and this can’t be changed; by the very definition of the word he hasn’t been reincarnated. A reverse reincarnation doesn’t cut it either because of the unchanged Gold Room sign - the audience is still in the same time frame. Jack's face was airbrushed onto someone else's body.
Stanley Kubrick does say the photo “suggests reincarnation” in the interview with Michel Ciment. but he cleverly doesn’t say where. The photo is obviously not The Overlook.
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