When it's online info, chances are the material has no peer review, so less trustful.
When it's paper printed, almost sure the material was peer reviewed, so chances are the book is correct, even when it defies common sense.
That's a good reason to search if there are Allen keys with different head profiles. Turned out square Allen keys exist:
https://duckduckgo.com/?q=square+allen+key&iax=images&ia=images
The definition was primarily of "Hex" followed by "(also called Allen)". So I'm sorry but as much as I appreciate your reasoning I cannot agree with you. In this case, Hex was used, as was Allen therefore I don't think it unreasonable to assume the item in question is a Hexagonal Allen. (Did I learn my shapes wrong in primary school?)
I am well aware of most of the different shapes that have been forged or machined onto the ends of allen
type keys , including mass produced and ones I have had to make myself for specialist applications.
I believe the name "Allen" being taken from the Allen Manufacturing Company (bought out with various setups since) who registered it as a trademark. The correct name for the tool made by any other manufacturer is
Hex/Hexagonal Key, but in the same way that we call vacuum cleaners hoovers, the same applies in this situation.
A square allen key is therefore not technically an allen for two reasons, just a plain old square drive key, and not even that would be called "squar
ish" (with the possible exception maybe a one off having to be made for a special situation).
I am more than happy to be educated if any of what I have just written is incorrect but submit this post as being correct to the best of my knowledge.
Never too late to learn new things, which is also what I am trying to do here and elsewhere with electronics.