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.
I mean it says that you can work anywhere. It does not explicitly say you can change completely to a new employer, but it also doesn't explicitly say the opposite.
.. I am not saying you are incorrect. But I personally interpret the text differently. Do you know for a fact from the Thai website that what you say is correct, or is it your "safe" interpretation?