“vacation” is what we use to describe taking time off of work here, and “paid time off” is specifically for describing how the time off is arranged with one’s employer.

This is an important note - PTO is for individualized, arranged time off. It could be used for a week, a single day, or even for just a few hours (in some cases.) It could be used to give yourself a random day off here or there, if you want to. It doesn't have to be a vacation or a holiday (in the American sense), just a random (though usually pre-requested) day where you feel like staying inside in pajamas, marathoning movies all day, and getting paid for it. (AKA a mental health day.)

It's also important (and sad) to note that not all jobs offer PTO. Ditto for sick leave and parental leave. The US is straight-up backwards...