It states that you need a letter addressed to the embassy and a certificate from employer/foundation/organisation.

Well, does this apply to public schools? If not will volunteering or even working at a demonstration school that is part of a university be a better option? I heard that the requirements for demonstration schools are less strict if they are in true need of help.

In all cases you need a letter from a reputable organization when applying for a non-imm 'O' for volunteering.

You will also need a work permit to legally perform said volunteering work in Thailand.

This from embassy website. http://vientiane.thaiembassy.org/vientiane/en/consular/consular_check/

"For the purpose of voluntary services, documents required are;

1. Recommendation letter addressed to the Embassy,coupled with the certified copy of the endorser's ID
or passport and work permit in Thailand
2. Copy of registration certificate of the employer/foundation/organisation
3. Copy of the previous/current work permit (if any) "

Certified in 1 means signed and dated by the endorser.

I don't think they get picky on what kind of school you are volunteering at to get the visa. That will matter when you apply for an extension of stay at immigration.