On lights, concur that a proper helmet mounted light is better than a headlamp for running or hiking. I experimented with my running headlamp while I was looking for a mounted light. While in my case, my Petzls (first my 6yo Reactik then its replacement Swift RL) worked ok fitting fairly neatly under the brim of the helmet and above my clear lenses, long term stability was a concern, as was control of the light.
I needed a light that could handle cold weather (down to -10 C), rain, and snow as most of my dark rides are before dawn, sometimes occurring entirely before dawn and ranging from 60-90min of complete darkness. I also didn’t want a light that required a permanent mount on the helmet nor require a battery in the jersey. For me, that meant the Lupine Piko R. It has a remote, is configurable, and is attached to my Kask Protune with a velcro strap without issue, jostling, or problems despite terrain.
BTW - the handlebar light was a Light & Motion Trail light, which was great in not-cold weather but proved unable to take the cold. Below around 5 C, its life was around 30min. I replaced it with a Bontrager Ion Pro, which besides being linked to my Garmin through the light network feature (so it turns on automatically at dusk, in tunnels, or off automatically at dawn), lasts nearly an 90min in sub-zero.
There are a lot of good options out there, just make sure the conditions you’ll be riding won’t kill the light. One of the helmet options I liked had virtually no water proofing and another showed no indication of handling cold weather.