That’s the key. Since you’re not a permanent resident, any contract you might have with an employer cannot exceed your permitted period of residence.
ETA: If you were a permanent resident (i.e. an APRC holder), since this (buxiban teaching) is a job that’s subject to the Labor Standards Act, and presumably your original contract was a normal one, you would presumably have an indefinite contract.
Also presumably, they haven’t needed to apply for a work permit each time you agreed to stay for another year, because as a foreign spouse you have open work rights. So while they have evidently failed to do their due diligence in confirming that they were following the law, they haven’t necessarily tried to deceive you.
Well not exactly. A contract does not need to be in writing (with some exceptions), so if OP agreed to continue working until X and did so properly (i.e. not working illegally), then it’s still a contract, though definitely not an indefinite contract in this case.
Because the Supreme Court’s decision on the impossibility of indefinite contracts for non-permanent residents has not (afaik) been codified in law, if the employer has trouble believing this, it’s probably best to ask a knowledgeable person at the local labor department to explain it to them.