Military and naval terms are many times interchangeable, such as the use of the term 'regiment' and 'battalion' in the British army of the period and during the War of the American Revolution.
However, what prompted my inquiry/correction was the following title:
A concise and accurate account of the proceedings of the squadron under the command of Rear Admiral Sir Will. Sidney Smith, K. C. , in effecting the escape, and escorting the royal family of Portugal to the Brazils, on the 29th of November, 1807, which was published in 1809.
The author of the above volume certainly makes a difference between 'fleet' and 'squadron' regardless of the explanation that you provided.
Further, the following definitions are provided from three naval dictionaries: (1) Nautical Dictrionary by Arthur Young, 1863; (2) A Military and Naval Dictionary by John Wiser and HC Gours; and (3) A Naval Encyclopedia from 1880 (no author, but it includes a listing of the contributors). All are available on Google Books.
Fleet: A general name given to the Royal Navy or any part of it employed in a particular service or expedition. No definition is given for 'squadron.'
Squadron: A division of a fleet or a small number of naval vessels acting together. In naval tactics the fleet is divided into squadrons-van, center, and rear and appropriate movements are assigned. No definition for fleet is given.
(a)-The whole naval force of a country.
(b)-A collection of ships, either war or merchant.
(c)-A fleet is divided into divisions and squadrons...
(d)-A small fleet is called a flotilla.
(a)-A detachment of vessels or boats employed on any particular service.
(b)-The division of a fleet...
(c)-Any assemblage of vessels smaller than a fleet.
So, it seems to me that Smith commanded a squadron or a flotilla, not a fleet, as a fleet indicates a large number of naval ships. Smith's command was a subdivision of a larger naval force, therefore it could be called either a flotilla or a squadron.
For example, the US and British naval forces on the lakes in the War of 1812 have been referred to from time to time as 'fleets.' However, in reality they were either flotillas or squadrons and were also referred to as such. I suggest that the latter two terms are appropriate. If all three were assembled together, which would be a relatively large number of ships on both sides, then they would be a 'fleet.'
Terming Smith's command a 'fleet' falsely inflates his command and importance and it should be remembered that he is a relatively small player both in naval circles and in the overall picture of the French Revolutionary and Napoleonic Wars.
I would also suggest that the importance of minor character should not be overlooked. I believe everyone has them for whatever period one is interested in-I certainly do. But their relative importance should be understood and by all means discussed and debated. However, inflating a minor character's importance into a major event or an overriding event is ahistorical and should be avoided at all costs.