It depends on how you define it, I guess but personally, I think of spin as something with the slow arm action and flighted trajectory, even if it is not that well flighted with drift.
Now, there are other ways to get "turn", and cut, as some of you have put it, is one way, where the effect is to get the ball to turn, same as "spin" but you can achieve it with a faster arm action and no real flight or drift.
If you see the slower ones of Bravo and Harshal Patel, and compare it to the regular offcutter a Cummins or a Bhuvi bowl, you will see the difference.
In my mind, spin and cut are two ways to achieve turn but they are not the same.
Your definition fits in the modern definition, yes, but a lot of it has nothing to do with spinning the ball. In Barnes' era, spin meant getting movement off the wicket, and where cut was mentioned it was a specific technique and wasn't a distinction always made. It's often used interchangeably with 'break', a term especially common in very old writing.
This is because especially before about 1900, where there was only one new ball (I think the seam was often pretty flat in those days too), all bowlers who expected to be effective got movement by some form of sidespin, whether the ball was spinning perpendicular to the pitch or just a little out of line. The softer pitches back then would have definitely favoured this type of bowling, especially in England.
Even later on it's common for bowlers far faster than any modern spin bowler to be be described as 'getting spin'. 'Flight' is also used much more broadly than just tossing it above the eyeline.
Here's a couple of examples:
Wisden on Ted McDonald in 1921 said:
It is scarcely an exaggeration to describe him as the best bowler of his type since Lockwood, combining as he does great speed with a fine command of length and very pronounced spin.
For anyone's reference, this is Ted McDonald:
Wisden on Chud Langton said:
With the new ball he made his deliveries swing late and rise awkwardly and when the shine had worn off he was almost equally a problem to batsmen because of his command of length, flight and spin, and also change of pace.
This is Langton, typical what was called 'fast-medium' at the time.
Going further back, Tom Richardson was considered genuinely fast yet plenty of reference is made to him getting a lot of spin. It's a usage that only completely dies out at the Second World War.
I should mention that with the predominance of spin grips were different too. The 'two up, one down' grip almost universal amongst modern fast bowlers was rare then and people held the ball all sorts of ways.
