Often "swords out, dice out" is a good rule-of-thumb. But whenever initiative questions arise, I prefer to look at the context, because I think MOST of the methods you describe could make sense depending on the specific context.
So my first step with your example is "reading the room." If it seems like the instigating player is alone in wanting to do this fight and the fight could have outcomes that disrupt other players' enjoyment, I'd pause, acknowledge the instigator's intent and readiness ("ok, you've gone for your sword"), and then I'd check in with the other players about what they want to do and what they'd like to see happen. This gives the other players a chance to talk down their instigating companion in a way that's non-disruptive and lets the instigator still take some spotlight in a roleplaying sense.
Similarly, if it's just this one PC and this one mouthy noble who are throwing down – the rest of the players are watching and just heckling, same with the noble's entourage – then in a 1v1 I'd dispense with initiative entirely and use simultaneous initiative.
However, if everyone's game for combat, then the instigator might go...
Normal Initiative Order: Everyone in both groups are clearly opposed, no player is going to try to talk the instigating player down, there's the opportunity/motivation for a fight, there are some stakes to the fight, it's not going to be interrupted immediately by the authorities, etc. Totally regular initiative order. If the player's intent is not to deal hit point damage but some other effect, like cutting off the mouthy noble's kerchief or something flavorful, basically it's just the player signaling "I'm picking a fight."
At the top of initiative: If I've narrated the opposition as particularly unprepared or overconfident. If the opposition was deliberately trying to lure the PCs into attacking first to get them in trouble with authorities or to invoke "self defense." If the opposition is overwhelming (i.e. no prayer for players to defeat them) and is toying with the PCs. If the opposition is really weak... and in this case initiative might not even be necessary because it's going to be over quick.
During normal initiative but with chance of people on both sides could be surprised: I would use surprise if the instigating player's intent is to get the drop on the mouthy noble's entourage; the player would roll Deception vs. the Insight of the enemies, and whether the rest of the PC's friends would need to roll Insight would depend on specific situation.
At the top of initiative, with the chance people on both sides could be surprised it's starting now: Getting automatic top of the initiative with surprise is something I rarely use as a reward for excellent recon/preparation/strategy/sabotage. Because your example scene seems more spontaneous, I wouldn't use this particular option for this context.
EDIT: I'm very much a contextual DM, and I prefer to play with players who can I can build trust with so I can creatively flex things like initiative to better reflect our shared story. However, I know some players hate this approach ("the rules are the rules"), so my step 1 is always read the group I'm running for.