In none of the cases described does a hard fork occur. It is a soft fork despite the forking of the chain. Both chains are valid Bitcoin blockchains but the majority of nodes are following the one that contains more work.

I suppose you then end up with some form of an alt coin, which could become Bitcoin in the future if the chain contained more work.

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I guess I misunderstood what a hard fork actually is. Point being, there would be two chains until the more restricted updated chain one outran the legacy one.