I think it is perfectly ok to boot, boot up, and reboot (restart) a computer. It doesn't need to be a personal computer, the term started with other computers.
The terms boot, comes from bootstrap. In the early days of computers it was a big deal to get the operating system software (operational software) into the device (computer hardware) to make it do something useful. You would manually enter a small program, that would allow the computer to load a larger program from some storage device. The small program was called a bootstrap loader. Eventually the small bootstrap loader was built into the computer hardware.
Yes, and the reason the small program was called a bootstrap was an allusion to the phrase "pulling yourself up by your own bootstraps", as far as I know. (I started work in the days when the bootstrap program was launched using a sequence of switches on the front of the computer, each switch represent one bit in a byte-long address. We've come a l-o-n-g way. )