A re-entry permit (multiple or single) does not care about calendar years. It is valid from the date you get it, and keeps your current admitted to stay date alive. It does not give you any extra time than you already have on your existing visa or extension of stay. A single re-entry is invalidated as soon as you leave the country and re-enter again. A multiple re-entry is valid for the entire period your current visa or extension is valid for.
On a non O A you have one year of validity on the visa.. So within that year you will get stamped in for a year every time you leave and re-enter the country. You can exit and re-enter just before the date of expiry to get another full year of stay. But if you leave and re-enter the country after the original visa has expired, you will need to buy a re-entry permit as to not cancel the permitted to stay date you already have. If you are already on an extension of stay, from immigration inside Thailand, then you always need to buy a re-entry permit to not cancel your extension.
that does not add up, when both families entered in August on a single entry that got extended.. The permitted to stay date is 4 months apart on the extensions.. Maybe the letters from the schools where different. Could be a typo in the letter, or a mistake from the IO. If the school terms is the same it has to be a mistake somewhere.