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Did this start after upgrading to macOS 16, or did it happen before?
Odd power issues can sometime be mitigated by resetting the System Management Controller, or "SMC". Apple instructions for the SMC reset are here:
Reset the SMC of your Mac - Apple Support
It's it important to use the instruction set labeled "Mac with Apple Silicon" for your M1 Mac, as well as note that the computer needs to be running on wall power, not the battery, when you reset.
If that does not change the behavior, then I fear that you have a defective battery that is reporting incorrectly, because of this:
My battery is showing 100% battery health and only has 131 cycles. It gets to about 60-70% and then shuts down and turns back on with 10%.
That health value is inconsistent with a healthy battery. It should not report 100% health at 131 cycles. Also a shutdown at mid-charge level followed by a low charge report at restart are just wrong.
Does System Settings > Battery > Battery Health show "normal"?
if the SMC reset does not correct the issue, I think you need to have Apple evaluate. When I had similar shutdown/restart issues, the battery proved faulty, requiring replacement.