That's pure nonsense
In 1907, President Theodore Roosevelt requested the U.S. Congress to create a new law enforcement agency in the Justice Department. When Congress opposed, Roosevelt created the Bureau of Investigation by executive order in 1908.
PDF direct download: (paragraph 1) https://www.researchgate.net/profile/Mathieu-Deflem-2/publication/331305857_Federal_Bureau_of_Investigation_FBI/links/5c76e5ae92851c6950466409/Federal-Bureau-of-Investigation-FBI.pdf
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Roosevelt and Bonaparte both were "Progressives." They shared the conviction that efficiency and expertise, not political connections, should determine who could best serve in government. Theodore Roosevelt became President of the United States in 1901; four years later, he appointed Bonaparte to be Attorney General. In 1908, Bonaparte applied that Progressive philosophy to the Department of Justice by creating a corps of Special Agents. It had neither a name nor an officially designated leader other than the Attorney General. Yet, these former detectives and Secret Service men were the forerunners of the FBI.
https://irp.fas.org/agency/doj/fbi/fbi_hist.htm
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......in 1908 President Roosevelt created the Bureau of Investigation by executive order and directed Attorney General Charles Bonaparte to develop the agency within the Department of Justice.
https://www.encyclopedia.com/law/legal-and-political-magazines/federal-bureau-investigation-history
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Moreover, Theodore Roosevelt and several others in his administration favored the expansion of the Secret Service to meet their desire for a comprehensive federal detective agency.Soon after Roosevelt issued an executive order creating the Bureau in 1908, however, the agency established a secure niche in the federal bureaucracy.
https://heinonline.org/HOL/LandingPage?handle=hein.journals/slps14&div=8&id=&page=
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Nobody denies that Theodore Roosevelt created the FBI. Some do later specify that he directed Bonaparte to go do this, go do that, go do the other thing, not that Bonaparte really needed all that much. Progressives see eye to eye on matters of taking a leak on the Constitution.
Most cites simply ignore the role of the executive order. Even more cites gloss over the original name of the FBI, because it such an insignificant trivia fact we might as well memorize how tall the building was and how many bricks they used to build the building.
The important part is that the FBI that harasses Donald Trump was created by President Teddy Roosevelt. It's joke that doesn't make anybody laugh that you trust the FBI's own web page without even going and attempting to look if there have been independent research on the matter. Which there have been. You also didn't take the time to scan the FBI's own pages to see if there were any contradictions there.
"There was no FBI under Teddy Roosevelt. So yes"
Even your own link says Theodore Roosevelt. So no, the FBI is #3 to your pitifully omitted list of two. And there's more than 3 BTW, I just wanted to focus here on this singular one. You also said:
"That first happened under Roosevelt in 1908."
A contradiction. You can't have it both ways not in 1908 and in 1908. This is severe Stalinesque hero worshipping cult of personality nonsense. It's embarrassing to watch you do this to yourself.
In this quote and others, you've selected out "facts" which suit your propaganda purposes, while ignoring any which don't.
Here is a summary of facts which matter:
There was only a minor shuffling of existing offices and a directive from Bonaparte to his DOJ attorney's to request investigators through Chief Examiner Finch.