If you're going to start with salary, at least don't tell them what you currently make, because that will instantly be their baseline. Just tell them what you desire and don't tell them that it is 20% or whatever more than what you currently make.
But really you shouldn't start with giving a number. They already have a huge advantage because they negotiate salary all day and you do it once every few years. Just ask them for the range and make sure the minimum is above your X+20% number. If it's not, let them know that their range is too low, and if they won't tell you, say, "thank you, next".
I don’t think this true if you are already near top of band (very lucky place to be in). Right now I have golden handcuffs and just ignore almost all recruiters because I know only a few select big companies (plus quant) can match my pay.
Startups are a bit more hit or miss though, and if they are small enough they probably won’t even have pay bands. You need to do more due diligence there
Does the baseline matter though? If I’m not switching for less than 1.2x does it matter if they know what $current pay is?
Either they can offer it or they can’t?
It still matters because of mental anchoring. In their mind they are being very generous by giving you 20% more than you make so their offer will be the minimum.
If they don’t know what you’re making now, they will think the minimum might be too little and if they really like you they will go higher than the 1.2x.
The question wasn't about whether one should go first here, the question was whether if I make $100k should say either
a) "I make $100k today and I want at least 20% more which is $120k"
vs
b) "I want at least $120k"
The difference there is a lot less obvious. Of course, in the second you might have a minuscule chance of them offering way over $120k whereas in a you would not.
They might be willing to offer 1.4x. If you tell them you want at least 1.2x, they're probably not doing 1.4x. If you ask them what the range is and they tell you a range, you can at least ask them for the top of the range or more. Worst case they just say they can't do X but can do Y, and then you have more info to negotiate with. Way better than not knowing and leaving money on the table.
But really you shouldn't start with giving a number. They already have a huge advantage because they negotiate salary all day and you do it once every few years. Just ask them for the range and make sure the minimum is above your X+20% number. If it's not, let them know that their range is too low, and if they won't tell you, say, "thank you, next".