This gets perilously close to ask a lawyer / ask an accountant, but the non-legal-advice version is "What motivates the question of whether a US-hosted website is 'an American business'?"
If a hypothetical entrepreneur is concerned that hosting with an American company might bring their data within the orbit of the American legal system, then that hypothetical entrepreneur should probably model American companies as routinely honoring lawful orders from the American legal system, and should probably model the American government as a first-world democracy which is very capable of getting a warrant for data hosted on domestic systems and, given that they have satisfied internal rationales to request a warrant or similar process, would feel substantially no additional barrier was raised by the owner of that data being a non-US person.
If a hypothetical entrepreneur is worried that hosting in the US will result in their business becoming a US-registered business, the hypothetical entrepreneur may feel happy to know that registering a business in the US is something that actually takes non-trivial work and that renting a server is neither necessary nor sufficient.
If a hypothetical entrepreneur is worried that hosting in the US might cause their non-US business or themselves personally to have income [effectively connected] with [a trade or business] within the United States that hypothetical entrepreneur should talk to an accountant and be prepared to give a very thorough accounting of the facts. Note that the hypothetical entrepreneur is not uniquely at risk there due to their hosting situation; there exist a wide variety of things which constitute a trade or business or could cause the American authorities to believe that a trade or business is being conducted in the United States.
I have taken the liberty of bracketing certain words in the above paragraph which are terms of art in tax administration, for your ease of searching for further information or discussing them with your professional advisors.
Please note that it is not the case that the US and a nation of your choice will necessarily agree that because a business is "an American business" for any of the above purposes that it is not simultaneously "a $NATION business" for the same or different purposes, in whole or in part.