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If there's only one switch then chances are you are right about the red wire being the "switched" wire. As you say, the black wire seems to be hot all the time. Don't connect the red to it unless that's the way it was originally wired.
Is there any other device or outlet in the area that does not work when the black wire is disconnected? I'd be surprised if it just stops there, but if all's well without it, then you could just cap it (after verifying that the red wire is the switched one for the fixture).
You are right to connect white to white, and bare to bare.