For a backpacking light, I look for three things.
1) Efficiency means packing fewer batteries. I remember vividly packing 8 AA for a single trip years ago. Now I pack one in the light and one as a backup in case bad stuff happens. The first battery will take me through a normal week long trip and the second is there in case I need to get out. Efficiency is much better than when I was a kid, but there still aren’t many lights that can do what I want on a single AA. Note that I’m expecting more that a candle light… 25-50 lumens for a trail (couple hours over a week) and 2-8 lumens in camp (couple hours each night) with a few times at 100+ lumens (omg, was that a bear!!) is expected usage.
2) Weight is another thing that has improved by leaps and bounds. Anything over a couple ounces is probably heavier than necessary.
3) Modes relates to efficiency because properly spaced modes allow you to use less light and energy when appropriate. I find 25 lumens is about right for walking over easy terrain and 50 works well at a brisk pace. A low level around 2 to 6 works well around camp and less than 1 lumen is fantastic in the tent. Any time I can get by on less than 25 lumens on am already efficient light, I extend my runtime significantly.
The headlamp I settled on when I researched this about a year ago is the zebralight h52. It’s a bit above your price range, but is a fantastic light and best in class for efficiency that I was able to find. It also has a great range of modes, good top end, is regulated and has an excellent ui.
If I was looking for a cheaper choice, I really like the l3 illumination l10c (l10 is a decent choice as well, but definitely different). The top end isn’t nearly as impressive and it’s not a headlamp, but the efficiency is very good and the modes are very well spaced. As a note, the clip is not great. It runs about $35 with shipping.