Edit: I’ve revised this post, but it’s even longer now. Scroll down to the summary near the bottom to save time.
That looks like something they wrote before PR’s were added to the new system (新制, what I call the LPA system). Remember, neither the LSA nor the LPA was amended for this; instead, they just did it through Art. 11 of the AREFP.
And Art. 12 of the AREFP makes the public school teachers’ retirement system apply to PR’s – but not the private school teachers’ retirement system. 
Oh, but wait. They did amend the LPA after all, last year.
(Skip explanation of the amendment to save time.)
That Art. 7 Par. 1 Subpar. 4 reference means
- Foreigners other than those mentioned in the preceding two subparagraphs who are permitted to reside permanently in accordance with the relevant provisions of the Immigration Act and are employed in Taiwan.
in other words, permanent residents.
So again:
-
normal (LSA) job + normal status (not PR or foreign spouse etc.) ==> LSA pension (old system)*
-
normal (LSA) job + PR/spouse status ==> LPA pension (new system)
-
public school 編制 job + PR status ==> AGRSBCTOSMPS pension
*but of course under the old (LSA) system, you don’t qualify for the pension unless you have X years with the same employer, which is why the new (LPA) system is probably better for most people.
But what we want to find out is…
-
public school 編制 job + normal status ==> ?
-
private school 編制 job + PR status ==> ?
-
public or private school non-編制 job ==> ?
The LIA (laobao) old-age benefit (老年給付) sometimes called a “pension” is completely separate, so you can qualify for it no matter which system you’re in, as long as you’re registered for laobao and reach retirement age.
As for (non-buxiban) teachers, the LIA has the following provision in Art. 6 (compulsory insurance):
4.Employees of government offices or public or private schools who are not legally entitled to join civil servants’ insurance or the insurance of teachers and employees of private schools;
The Chinese version says you only need to be a 員工 (staff member) who doesn’t qualify for teachers’ insurance. It doesn’t say 編制, so if I’m not mistaken it applies to all employees who are not 編制.
And also in Art. 76 it permits you to transfer from laobao to private school teachers’ insurance (or military or public servants’ insurance) without losing your seniority for 老年給付 purposes.
The Teacher’s Act (which applies to both public and private schools) does state in Art. 31 that teachers are entitled inter alia
- To enjoy rights to and protection of remuneration, benefits, retirement, bereavement compensation, severance with pay, and insurance.
And yet, the same law states in Art. 38 that
Teachers’ retirement, bereavement compensation, resignation, severance with pay, and insurance is governed by separate legislation.
So where would this legislation be? 
(Not in the Statute Governing the Retirement of School Faculty and Staff, as that was repealed in 2019.)
For public school teachers it’s easy enough to find the retirement part:
As we’ve seen, this is only for 編制 (establishment staffing, staffing complement, manning strength… it would be nice if they would pick one translation and stick with it) teachers, i.e. full-time and qualified.
NB Art. 7 allows a former private school teacher who now works at a public school to use the private school years of service towards seniority for retirement, severance and bereavement, under certain conditions. The 編制 caveat also applies to the private school years.
As for the insurance part:
The Public School Teachers’ Insurance Act (no official translation available) includes inter alia the following as the objects of so-called 公教人員保險:
二、公立學校編制內之有給專任教職員。
編制 teachers at public schools
三、依私立學校法規定，辦妥財團法人登記，並經主管教育行政機關核准立案之私立學校編制內之有給專任教職員。
編制 teachers at private schools
and
四、其他經本保險主管機關認定之人員。
anyone else the competent authority (which is actually part of the Examination Yuan 
in this case) deems worthy.
What about non-編制 teachers?
First of all, about private school teachers’ insurance outside the scope of the above:
That’s right(?). They have a whole banfa just for transferring from laobao to military / public servants’ / private school teachers’ insurance, but apparently private school teachers’ insurance no longer exists, the relevant law having been repealed on 1999-05-29, presumably because it got merged with “public” school teachers’ insurance. What’s even funnier is that the banfa was last amended on 2008-12-31 without removing reference to it.
But at the very least, public/private school teachers who have no teachers’ insurance should automatically qualify for laobao (which isn’t much but is something).
As for private school teachers’ pensions…
If we go back to the Private School Law, we find this in Art. 64:
Matters relating to retirement, compensation, resignation and severance of employees of school legal persons and teachers and staff members of their schools shall be handled by regulations separately drawn up.
There’s more, but that’s the important part. The translation is a bit off as usual.
學校法人及其所屬私立學校教職員工之退休、撫卹、離職、資遣等事項，另以法律定之。
It’s not a separate regulation but a separate law.
And that law is, as Tando pointed out, the Act Governing the Retirement, Bereavement Compensation, Discharge with Severance Pay Benefits for the Teaching and Other Staff of School Legal Persons and their Respective Private School(s).
It also allows you to count years spent teaching at a public school towards your private school pension, but again, you need to be 編制.
We should also note Art. 39:
The wording of Par. 1 Subpar. 4 (外籍人士擔任各級已立案私立學校編制內專任合格有給教師及前三款人員) is ambiguous. You don’t need to be 編制 if you have one of the jobs in Subpar. 1 to 3 – unless you’re foreign? Or the mention of 編制 there is superfluous because it’s implied anyway in the mutatis mutandis (going back to Art. 3)?
If we go back to the analogous law for public school teachers, there is a similar provision in Art. 94 Par. 1:
With the exception of the provisions of Article 15, Paragraph 2, the provisions of this Act apply, mutatis mutandis, to the retirement, bereavement compensation, severance pay, and resignation-related refunds for professional staff members of public social education institutions, and research personnel at academic research institutions subordinate to the associated competent authority.
And then they mention foreigners in Art. 95:
Why bother mentioning in the AREFP that PR’s with 編制 public school jobs are subject to the public school teachers’ pension system, when according to the law for that system they don’t even need to be PR’s? Was Art. 12 considered necessary to stop them from having two pensions (LPA pension + public school pension)?
Answer: apparently yes.
But then why make no mention in the AREFP that foreigners with 編制 private school jobs are subject to the private school teachers’ pension system? Does that mean they do get two pensions if they’re also PR’s? 
Answer: no, because Art. 7 of the LPA prevents that.
Anyway, to sum up, for insurance:
-
any job that qualifies for compulsory labor insurance (laobao) – which should include non-編制 jobs at public and private schools – or voluntary labor insurance that the BLI approves ==> LIA old-age benefit (老年給付)
-
public school 編制 job ==> public school teachers’ insurance old-age benefit (養老給付)
-
private school 編制 job ==> also public school teachers’ insurance old-age benefit (養老給付)
And to sum up for pensions:
-
normal (LSA) job + normal status ==> LSA pension (old system)
-
normal (LSA) job + PR/spouse status ==> LPA pension (new system)
-
public school 編制 job + any legal status ==> AGRSBCTOSMPS pension (public school system)
-
private school 編制 job + any legal status ==> AGRBCDSPBTOSSLPRPS pension (private school system)
-
private school job that falls under the LSA ==> see above for normal (LSA) jobs
And, unless I’m missing something…
- public or private school job that is neither 編制 nor LSA ==> black hole of pensionlessness
Commentary:
The exclusion of non-編制 teachers at both public and private schools is in my opinion a form of discrimination against the precariat, which – based on the finding by the Executive Yuan’s human rights committee ~10 years ago that the clause in the LIA making laobao non-compulsory for employees of small companies (fewer than 5 employees) was incompatible with the international human rights standards that Taiwan officially incorporated into domestic law around that time – should also be changed, as it needlessly excludes some hard-working people. It could be changed by adding precarious teachers to the LSA/LPA pension system, by adding them to the same pension system as 編制 teachers, or by creating a new system.
That finding by the EY still hasn’t resulted in an amendment, so don’t hold your breath. 
There is a human rights committee at that suddenly-in-the-news-again institution (the CY), in case anyone wants to give it a shot.
Again, my apologies in case I’ve missed anything.