If the client includes an Accept: header in the API request, then the template path is not the same as if it is not included

POST https://.../match/api/1/v1/people/.../... HTTP/1.1
=> Response Template = /app/src/Template/TeirApi/response.ctp

POST https://.../match/api/1/v1/people/.../... HTTP/1.1
Accept: application/json
=> Response Template = /app/src/Template/TeirApi/json/response.ctp

Workaround by copying or symlinking the file in the other location.