Do you work fixed price or day rate?

Fixed price per phase, almost always. The reason is about where the risk sits.

What each one does

Day rate puts the risk on you. If it takes twice as long, you pay twice as much. That is fine if you have your own technical people managing it closely. If you do not, you are funding an open-ended commitment and hoping.

Fixed price puts the risk on me. If I estimated badly, that is my problem, and I have absorbed it before. It costs a bit more up front because I price in uncertainty, and you should expect that.

Why fixed works for the kind of clients I have

Most of them are running a business, not a software department. They need to know the number to make the decision. "Somewhere between twenty and sixty thousand" is not a number you can take to a board.

What makes fixed price possible

Discovery. I cannot fix a price on a description — I have tried, and it went badly. I can fix a price once I have seen the work happen and know where the exceptions are.

If someone offers you a fixed price with no discovery, they have either padded it heavily or they will be back for more later.

Where I do use day rate

Genuinely exploratory work, small ad-hoc changes after launch, and cases where a client wants to direct things themselves hour by hour. All fine — just named as such rather than dressed as a project.