It was a formal lease as I was staying there for 2 months. But I would think they're one in the same. They just need a real looking, signed, addressed paper in their hands.
I opened one on Visa Exempt with just copy of lease + passport. I tried 3 banks, 3rd one was a charm - Krungsri @ Central.
Before you go the visa agent route - just spend a day going to as many locations as possible. Expect to hear ‘No’ many times but eventually someone will say yea. Make a day out of it.
Also maybe try non-foreigner locations like in Thai neighborhoods etc.
As it was explained to me - If you’re not leaving the country and applying for additional visas and visa was granted by immigration, then ‘Extension of Stays’ are synonymous with getting the Non B