So, if I understand you, if you utter a 'threat' (scare quotes) to conditionally punish, but you would have conditionally punished even if you had not uttered, then it isn't really a threat, it is simply a 'warning'. Is that what you meant?

Or, that it is the kind of threat that someone could give to me and call 'pointing out consequences' without me holding them in contempt as well as antipathy as a result.

I don't think I agree with that characterization of the threat/warning distinction. Many people would not punish if they had not given 'fair warning'. But by your definition, such 'fair warning' is actually a threat.

I wouldn't use that definition and especially wouldn't use that distinction both ways. ie. I wouldn't actively assert that 'threaten' and 'inform of consequences' are mutually exclusive. I speak primarily of what would make talk of 'inform of consequences' appear an obnoxious attempt at condescension.

I'm not sure what you meant by 'status', 'institution', and 'sincerity', so they may constitute a piece of the puzzle. My own intuition is that part of the distinction lies in whether the prospective punisher has the 'right' to punish.

Yes, I mean approximately the same thing but with slightly more reduction.

But I don't have a good handle on whether the prospective punishee needs to deny the punisher's right to make it a threat, or he also has to deny the sincerity of the punisher's claim to the right.

I suppose it has to be the mind of the threatener that matters. I've had people threaten me with things that I actually consider desirable. (Failure of other anti-optimization?) It would feel disingenuous of me to declare that what they are not doing is not threatening. Of course if I could make the disingenuousity witty I would take pleasure in uttering it - people who threaten me don't have ethical rights to things like courtesy as far as I'm concerned!