That's not really true, the truth is that the distinction "spinner" or "fast bowler" didn't actually exist back then, the general distinction was Slow bowler, Medium bowler and Fast bowler. a slow bowler was a Hedley Verity, a Medium howler was a Sydney Barnes and a Fast bowler was a Harold Larwood.
The term "break" or "spin" was used for any and all deviation off the pitch, when a pacer did it, it was called cut but both the term spin and break were completely respectable for medium pacers, this is why there is such confusion because the term "spin" is used entirely by modern fans and Cricketers for the slow style of bowling but back then, "spin" or "break" or "cut" all meant the same thing, deviation from the pitch
Applying modern distinction is generally hard but his "breaks" were cuts. though Cut and Wrist spin are identical, just one has more rip to it and other relies on seam.
"Barnes' "leg-break" was "cut rather than spun". – Sir Pelham Warner, Barnes' Captain
"I also had the experience of facing Sydney Barnes when I was in the Yorkshire Second XI and he was still playing for Staffordshire in his late fifties. I made enough runs to understand why many will have it that he was the greatest bowler of all time. Even then he cut and swung the ball and used the crease brilliantly. Before he stripped to play, he used so many bandages and elastic supports that he might have been a mobile Egyptian mummy, and when he retired in 1934, at the age of sixty-one, his wickets still cost only 11 runs apiece." – Sir Len Hutton, The Greatest English Captain since the second war.
You have too proper sources who faced and saw him saying his "breaks" were "cutters", and he himself said he bowled as a fast pace (IE much quicker than spinners), and we have accounts of him swinging the ball.
Here are accounts of him bowling in-swingers
“I played three different balls. Three balls to play in a split second- a straight ‘un, an in-swinger and a break back ! Then along came one which was straight half-way, not more than medium pace. Then it swerved to my legs, perfect for tickling around the corner for a single. But the ruddy thing again broke across after pitching, quick off the ground and took my off stump!” – Clement Hill, the First Great Australian Batsman.
Here is an outswinger, leg to off IE from the leg stump to off stump, Swerve = Swing, basically an outswinger.
“On his great Australian tour he clean-bowled Victor Trumper at the height of his powers, a ball swerving from the leg stump to the off and then breaking back to hit the leg. It was the sort of ball, that a man might see when he was tight. I was at the other end, I should know.” – Charlie Macartney, Great Australian Batsman.
Here is confirmation he bowled both, outswing and inswing, and both were late.
“On a perfect wicket Barnes could swing the new ball in and out very late, could spin from the ground, pitch on the leg stump and miss the off.” – Clement Hill, The First Great Australian Batsman.
Here is one final description, saying his pace was more than medium Fast, he swung it in late (late inswing) and then had it straighten after pitching (leg cutter), that's the main Barnes ball. Also confirms he had both outswing and inswing and both off cutter and leg cutter but that much was obvious
""At appreciably more than medium pace he could, even in the finest weather and on the truest wickets in Australia, both swing and break the ball from off or leg. Most deadly of all was the ball which he would deliver from rather wide on the crease, move in with a late swerve the width of the wicket, and then straighten back off the ground to hit the off stump". – Harry Altham, Ex-Cricketer turned Historian, lived through Barnes' era
Basically Barnes relied on late swing both ways, huge leg cutters, bounce and so forth to get his wickets, and was directly put in the same distinction as Maurice Tate and Alec Bedser, clear medium pacers. He's just a medium pacer, that's all there is to it
