Those aren't the only two options. In fact: they are both bad options!
Here's a much better option than either. I'm sure it's not the best way either, but the fact that "what came off the top of my business partner's head" is so much better than hourly or fixed is a good indication of how bad hourly and fixed are.
Charging by the day is better than charging by the hour, and week is better than day.
Better than each is to stop valuing your time as "just" time. You'll never have those moments again.
Value your work product. Don't value your work effort, and definitely not your time.
Actually, don't value your work product. Figure out how your customer values your work product and charge that price.
Once you start to realize that your time is the most valuable thing you'll ever have and simultaneously completely worthless as a unit of currency, you'll begin to trade with what you have that is truly of value to the person actually paying the bill: that thing you haven't created yet.
I was going to bookmark the linked comment, and found that I had in fact already done so. It is a very useful, actionable piece of advice. Thanks for writing it.
I bill by the week. At the time, my agency rate on this project was $10k a week, which included the full time attention of a senior developer (~4 full days), plus part time Q&A and PM oversight.
Here's a much better option than either. I'm sure it's not the best way either, but the fact that "what came off the top of my business partner's head" is so much better than hourly or fixed is a good indication of how bad hourly and fixed are.
https://news.ycombinator.com/item?id=7850335