Nothing to do with where the earnings come from, it is all to do with where YOU are when you complete the work. Maybe a little research on your part would be advisable, and then you too would know that it was illegal. (Not gonna comment on why I am a mod). But, see you in a few days. :-)
"I tried extending down there, but they wouldn't accept my salary plus earnings from online teaching.", you tried to show them earnings from illegal work and they would not accept it, not really a big surprise there.
You could be asked for proof of onward travel by your air carrier, and immigration could ask you to show 20,000 baht or equivalent and proof of accommodation.
You will require the following steps to be completed; It is mandatory that the following steps are taken before the Non-Immigrant “ED” visa for the Monk can be issued:
1. Recommendation letter from the temple in Thailand sent to the Office of Buddhism, Thailand.
2. Recommendation letter from the Office of Buddhism, Thailand, sent to the Ministry of Foreign Affairs, Thailand.
3. Approval letter from the Ministry of Foreign Affairs, Thailand, sent to the Royal Thai Embassy.
4. The Royal Thai Embassy issues Non-Immigrant “ED” visa for the Monk (visa applicant)
Guess you are talking about the ETA, which is not even going to be "soft" introduced until December this year, so in answer to your question, "no" you will not require one as they will not even be available at the time of your travel.