Lithium ions are safe when used cautiously, especially protected cells which you mentioned. These have a built-in circuit which keep them from being overcharged or over discharged.
Your main two options for batteries are AA size versus 18650. The AAs are eneloops as mentioned above although you can also use lithium ion batteries which are AA sized called “14500”. Eneloops are completely safe and great batteries. 18650 will be heavier, larger, have much more power to give you a brighter and longer lasting light.
The first two digits in the numbered battery give the diameter of the battery. The last three give the length of the battery, both in millimeters.
I am a big fan of AA headlamps as they are light and usually provide more than adequate light. I like to pair them with an 18650 hand held light which provides more throw.
Here is a nice AA sized headlamp that is good quality and would use eneloops or 14500s.
https://www.amazon.com/ThruNite-Neutral-White-Lumen-Headlamp/dp/B01FCV32SI?th=1&psc=1
You want the “neutral white” color.
There are many other options although many of them in that price range will be plastic where is the thrunight is extremely well-built aluminum without being excessively heavy. There are many other good options. Others will be along with suggestions here I am sure.
Eneloops can be purchased on Amazon as well. There is a regular version as well as a “pro“ version. The pro version has more power although won’t last as long in terms of charge/discharge cycles (perhaps 500 cycles versus 2000 cycles). Most folks are fairly happy with the regular version. if you have an IKEA nearby, you can also buy “LADDA“ cells which are very similar/identical to the eneloop pros but are cheaper.
There are also many options for chargers. You can spend significant mounts of money. I am a fan of the. Liitokala 202 for cheap yet reasonably good charger. It can likely be found on eBay for a few dollars more.
https://m.gearbest.com/chargers/pp_424662.html?lkid=11963380
Try to stay away from headlamps that use three AAA batteries. These usually do not have a circuit to maintain the lights brightness as the voltage drops. Headlamps with a single-AA will have such a circuit and give more consistent light As the battery drains.