Can I register as a tax resident in Thailand while on a tourist visa?

That is correct. To get a tax ID last I heard they advise to open a local bank account which will give you one automatically. Thailand elite visa includes a bank account.

Once you do and you spend 180 days per year in Thailand, you qualify as tax resident, which means you don't pay any taxes on income from a foreign source. From what I've read this isn't a foreign vs local sourced distinction as in most other countries (as in in which country were you when you worked for that money), but rather if the money enters Thailand or not, which is a huge advantage (most other countries with territorial tax systems do tax locally sourced income, even if it doesn't enter the country).

The bigger issue is your country of origin and the country were you earned that income, because both will be interested to tax you. Many countries have some unclear rules you need to comply with before they consider you as having moved to another country (which can be having a long term residency permit, spending minimum 2 to 5 years in said country, not having any attachments in your country of origin...).