"DOES NOT ENGAGE IN ANY BUSINESS OR PROFESSION WITHOUT THE CONSENT OF THE DIRECTOR OF IMMIGRATION"
Pretty poorly written or at the least ambiguous. If taken as written, it could mean anyone who was in business or employed could not vacation in Belize. Should probably say "does not engage in any business or profession in Belize".
I am sure that is the intent. That is why my interpretation concerning working via the internet, would be that if a person was doing business outside of Belize and was not a citizen or permanent resident of Belize, they would not be liable for any Belize income tax.
In the US a citizen is liable for income tax on income generated anywhere in the world. A person residing in the US that is a citizen of another country is only liable for income tax on money earned from business in the US or from a US company.