IF you have an account with Bangkok Bank, TransferWise transactions are shown on your statements as international transfers. Bangkok Bank are a clearing bank so if your account is NOT with them (as TransferWise apparently use them to clear funds into Thailand) these transfers may well NOT show as coming from abroad and you may well have problems demonstrating that to Thai immigration should you need to.
If you want to volunteer at a local school you should get the relevant police clearances. The appropriate visa and the correct work permit.
The school would need to support you in that, and they probably won’t.
You can think it should be however you like, but it isn’t like that it’s how it is.
Volunteering on a visa that you won’t get issued a work permit on, like the Non O-A you’re planning to get will leave you open to all kinds of potential problems. The main root of all those problems is that it’s illegal. You’re not a citizen of Thailand. You can’t do the things that a citizen can do. Once you get your visa you will be an alien who has been permitted to stay here under the terms of the visa you obtained. Aliens require work permits in order to work. Volunteering is regarded as work. It is work. It’s unpaid work but it’s work. Work that aliens require a permit for. You won’t get one so if you do that (unpaid) work you’ll be doing it illegally. This is something you need to decide if you’re comfortable with or not.
Personally, when arriving on a 30 day exempt basis, if I’m asked how long I’m staying I say 30 days. It’s just easier. They get to stamp me in without saying anything. I get to go through immigration without talking to them. Everyone’s happy.