Interesting that some require a certificate. I always assumed they didn't insist on any qualifications except a degree of some sort so they can get a visa.
Most places want to see you know about grammar and "theory" these days. Job vacancies request many documents and degree's are just one of a handful most companies require. Here's the list:
Passport (obviously)
Valid Residence Card or Certificate of Eligibility (CoE will be arranged by your company if you're still in your home country)
Teaching Certificate (Depending on the job, you might need a recognised classroom / full teaching certificate from your home country, especially for Homeroom Teach jobs)
University degree (generally a mandatory requirement and these days, since degree's mean nothing, even an idiot with a BA (Hons) in Flower Arranging ca get themselves to Japan whether or not they can spell their own name and used Grammarly for their dissertation.)
120 hour+ TEFL Certificate (a job that doesn't ask for this is rare but they may be satisfied with your degree IF it's an English degree.)
Early Years and Young Leaners Certificate (not always but this can help with Kinder jobs)
CELTA / DELTA (Not always and these are not guaranteed to help negotiate for a better salary either)
Drivers License (becoming more common with certain jobs)
Those in bold are generally what are asked for...insisted on.