Non-Immigrant status
Usually the clearest starting point, whether the underlying reason is employment, retirement, a Thai spouse or dependant, study or another qualifying long-stay route. What the bank then asks for depends on that reason rather than on the visa label alone.
Work permit holders
Some banks publish a work permit as part of their foreign-customer requirements for a standard savings account. Where that is the published rule for a product, it applies to that product at that bank — not to Thailand as a whole.
Tourist visa or visa exemption
Do not plan around it. Ordinary account opening under tourist-type status should not be treated as available, and building a relocation timeline on the assumption that it will work is a poor bet.
Destination Thailand Visa (DTV)
The DTV is a long-validity visa, and that is exactly where the confusion starts. A long validity does not automatically mean a bank treats the status as suitable for ordinary account opening. Some major banks currently exclude DTV holders from their standard foreigner savings products.