You don't have a maximum salary that you want, but you have a minimum.
The company has both a minimum and a maximum. I argue that when a candiate's ask is too low this is a flag that something is wrong with them.
So there are 3 possible scenarios with asymetric information; nobody knows the true scenario from the beginning of the negotiation:
1) Company Min <= Your Min <= Company Max
This means you can work here. Whoever names a number first is likely to form a ceiling or a floor, but both parties are aware of that fact. It's just a question of who goes first.
2) Your Min <= Company Min <= Company Max
Typically happens for underqualified candidates.
You can work here if you fool them. The only way for that to happen is if you get them to name their minimum (or maximum, and infer the minimum from it) and realize that your ask is too low. If you name your true Min first, you lose the 'game'.
3) Company Min <= Company Max <= Your Min
You can never work here. The best thing is for you to say "My expectation is X" and they will say "Sorry, we can't do that", but at this point the ball is in your court to accept a potential counter-offer.
There is a number of over-simplifications here but: in your original question you assume that you land in #1. I just want to illustrate that this may not be the case. And if it's not, you DO want to name the salary you want first but you want to lie about your actual expectation to make the number higher.
Ceteris paribus, from the company's perspective a confident candidate that asks for a very high rate right away (#3) will trigger a reaction of "Damn, probably good but we can't afford her." Like the people who negotiate cars down from artificially high prices, they will feel good in the tommy if they manage to get that candidate down to a lower rate and hire her.
I advise to name your expected salary first, but err on the side of being too expensive. They will never outright reject you if you are worth your beans. If they like you, they will counter-offer, but you will never end up being under-paid and sometimes you will be pleasantly surprised when your higher ask is met. The downside is that you will miss out on some lower-paid job opportunities - I think we can all live with that.