This is a complex question and the precise specialism I have worked decades on (in EU) unfortunately I dont believe you will get a clear answer as the choices involved are dependant on the individual tax treaty, the host country, and are often so vague as to become political as much as legal. Essentially if they want to, they can, is very close to the reality. This is so grey an area even mega money international tax advisors will shy from putting their name on all clear statements.
In the EU they can define a PE from extremely broad circumstances, I lost a >10 million euro company to the Dutch Belastindienst at the end of the 90s because I used a Dutch fax line to gather some timesheets, as the clients site foremen didnt like sending them to my head office on a international dialled line !! As simple as receiving a fax or phone call gave them enough ammunition to call it a domestic permanent establishment under OECD style legal interpretations.
The chance of it being an issue is near zero but not actually zero. Thats what HQ corp lawyers are likely flagging.
To add, I wrote my own letter on my own headed paper and it was fine.
But I also have years of filed corp accounts, employee personal tax returns (you have to be an employee and draw a min wage salary as Dir in the location my corp is based), payslips, turnover in the company, etc.
People legally really earning online as a remote worker should have some of the above.
You might get away with writing your own but they are getting tighter on non earning 'workers'.. Just show them you make a real income online.
You have a tax return on income made ?
You have filed accounts for your LLC ?
Your LLC must have clients ? So get a letter from them ? Or show payments from platforms.
Basically show you have a real income earning business.. Not just formed a 100 USD LLC and applied. The check they are making is being a self supporting remote worker, its not just being the owner of a corp formation anyone can buy for pennies, that doesnt actually support your lifestyle.
Tourist visas are tourist visas !! DTV are Destination Thailand Visas.. There is no catch all for tourist visa in the same way there are for non immigrant visas.