If you change the name of the status under Admin -> Issues -> Statuses, then yes it will affect the way it appears on the other workflows.
If you go to add a status to your workflow, (<Workflow> -> Edit -> Add Status) and you start typing the name of an existing status, you will be able to use it in your workflow. All of the Transitions to and from, validators, conditions, and post-functions will be specific to your workflow and will not affect other workflows that may use the same status.
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