The best under-$100 battery powered headlight I've heard of -- per experienced users and web reviews with light pattern tests -- is the
Busch & Muller Ixon IQ Premium. Takes four AA batteries, alkaline or NiMH rechargeables. Great, evenly distributed beam pattern to light up the road without blinding other cyclists or motorists. And the price has recently decreased about $30, so even with international customs and sales taxes, the Ixon IQ Premium, light shade, and fork mount, will cost less than $80.
You can find YouTube demo reviews by ordinary users for the B+M Ixon IQ and IQ Premium compared with other lights and it's clearly the best in its price class.
I'd planned all year to buy the Ixon IQ or Premium myself, but a series of nagging costs for repairs, higher medical costs for prescriptions, etc., kept putting that purchase out of reach. So I finally settled on a
Serfas SL-255 for $35, total, via Amazon Prime. Bright enough for safe commuting, runs on a pair of AA batteries, more than half the light for half the price of the excellent B&M lights. I've had a few night rides with the Serfas SL-255 and it'll do nicely. Subjectively it's as bright as car headlights on low, but with a narrow round (oval on the road ahead) pool of bright light surrounded by a weaker wider rectangular spill that helps illuminate toward the sides a bit.
I can't think of a USB rechargeable headlight I'd recommend because none of them ran long enough per charge on steady bright to suit my needs. I often begin rides early in the day and don't get home until midnight or later, depending on social occasions, visits to museums and galleries, etc. I wanted a headlight that would run at least four hours continuously at maximum brightness, and needed only a quick battery swap to keep going. I can carry spare AA batteries, or if necessary buy 'em from any late night store.
I have two or three red LED taillights that are adequate and run off USB rechargeables or CR2032 button cells, but I'm looking for another brighter taillight to bolt onto the rear rack that runs off AA batteries and will last much longer per charge.