1. The Institutional Split: After having my Taiwan-registered attorney colleagues check the rules and then call over again to confirm with the Council of Labor Affairs and again with the Taipei City authorities, they reached the conclusion that private buxiban teachers are subject to the Labor Standards Act. There's apparently a lot of confusion on this issue because there are a lot of teacher-like jobs that fall under the Teacher's Act (教師法), which is helpfully available in English and Chinese at the MOJ website. Article 3 of the Teacher's Act perhaps leads into some of that confusion because it states:
This law applies to full-time certified teachers with monthly salaries in public and registered private schools.
Perhaps it's natural many would assume that because private buxibans also have some sort of "registration" from the government that the teachers in them would fall under this law. However, the CLA, as the competent authority, gets to make the final call on how they want to divide up the types of schools and who is covered by the Teacher's Act and who is covered by the Labor Standards Act. And they do make a split at private buxibans.
The line drawn by the CLA is that the Teacher's Act applies to all the teachers at: elementary schools, middle schools, high schools, vocational training centers, colleges/universities. For background on this, the government came up with the rule because of the difficulties of calculating teachers' working hours, overtime, etc. -- there's an economic reason for their putting many school teachers under the category in that there was some fears of overtime obligations. The private buxiban teachers, according to the CLA, are covered by the Labor Standards Act -- perhaps because the nature of their work is simply thought to be different and "more contractual" or "paid by the hour" than the sort-of extended grading, traffic-control, etc. work that a teacher might be expected to do without regard to overtime pay in one of the other sorts of schools.
2. The Job-Specific Split: Also note that the split goes by what the employee does at an institution that falls under either of the two laws. Thus, a teacher at a public or private high school, middle school, elementary school, etc. might be under the Teacher's Act, but all the non-teacher sorts in the same institution would be under the Labor Standards Act. For example, Groundskeeper Willie, were he to find a job as a groundskeeper at a Taiwanese public or private school covered under the Teacher's Act would still have his employment rights covered under the Labor Standards Act.
A Quick Sidenote: Currently, certain aspects of Taiwan's employment law fall under the Pension Act only if the employee is a Taiwanese citizen. Foreigners don't yet get to participate in the Pension Act scheme for retirement, and the stingier provisions for severance payment calculations therein also do not apply. Thus, severance for foreigners who fall under the Labor Standards Act is going to fall under those LSA rules.