You could use both statements during a typical ideation phase but "As a user-statements are typically used later in the process when you want to document your ideas.
There are a lot of branded processes out there that promise that if you follow Process X or Y everything will be great. It could be a daunting task trying to understand them all.
But, in general, all creative processes, the loose ones, and the strict ones usually go through something like this:
- Problem - The design problem is defined (the problem can be defined by a client or by yourself. Something needs to be solved. This could sometimes be superspecific like "The login-screen doesn't work" or "Improve the brand attitude". Some people call this "The Design Challange" Purpose or whatever.
- Research - Why doesn't the login-screen work? Why don't people like our brand? You might do user research, market research, look at successful competitors, or just analyze the problem based on your own experience. Some people call this Discovery, Inspiration, or whatever. It's the same.
- Ideation - This is the phase where you can start to think about how to solve the problem. Now, some organizations or authors want to be effective and try to streamline the ideation phase, so they come up with different methods and techniques, such as "How might we", "As a user..." Job stories, Brainstorming, Brainwriting, Frame storming, Kick start questions, Focus questions, etc. There are tons of methods out there that all promise to be the most effective one.
"How might we" is an ideation method developed by the design agency IDEO and Stanford University. The phrasing is meant to open up the possibilities. If there is a problem with the login-screen - do we really need one in the first place?
"As a user" is tied to Agile processes. You could phrase a statement like "As a user I want to login as fast as possible." and then brainstorm ideas around that. The problem with using an "As a user" statements is that they could limit the possible solutions to the problem. With a statement like the above, you would likely focus your ideas mostly on users and performance.
This is also the phase where you also develop sketches and prototypes.If you are into Human-Centered Design you want to test these prototypes on users or just show them to smart people that could help you improve the design. Some process authors divide the ideation phase and the prototype phase (using labels such as prototypes, Build, Learn etc)
Feel free to experiment and find the techniques that work for you and your team, if you have one. Just thinking about a solution to a problem could work wonders too. You don't have to use any of the methods. Plain thinking is a method too.
- Execution - This phase is all about actually building the thing that will solve the problem. In a typical agile process, you document what needs to be built by using statements such as "As a user". But if you are a small team and are able to just speak to each other you don't need As a user-statements.
Now, this all seems super structured and super linear but in the reality you could start anywhere in the process. You could start with, an idea and research to see if there is a problem to be solved. You could start with research to find a problem. You could start with the problem, move to the idea-phase and do research to see if the idea could be improved. It's a mess but that is what creativity is all about.