You're Correct!

Any golf shot which is struck cleanly and rises into the air will have back spin – period!
When the ball moves either left or right in the air its not side spin but a tilt in the back spin axis which causes the deviation in flight.

A dead straight shot occurs when the club moves into impact on a straight path with the club face at right angles to the target line. If these two factors are correct and the ball is struck from the centre of the face, the clubs loft will cause the ball to spin vertically backwards around its own axis flying upward and straight. The spin axis will become tilted by different combinations of face and path. For example, to hit a fade the club needs to travel from out-to-in (cutting across the body). At the point of impact, the club face needs to aim between the target line and club path (just left of the target). This will produce fade spin axis and move the ball left to right.

For a draw, the club needs to travel from in-to-out (moving from inside the body to out). At the point of impact, the club face needs to aim between the target line and club path (just right of the target). This impact position will cause a draw moving the ball right to left.
Slices and hooks occur when the club face angles become more extreme and tilt the back spin axis even more.

One of the most overlooked aspects of the ball moving during flight is where on the club face it is struck from. Toe and heel strikes can cause the ball to move through the air as the face bulge comes into play. The face bulge on a club, especially the driver, causes the gear effect to take place. A shot struck from the toe will generally cause the ball to move from right to left in the air whilst a heel strike would cause a shot to go left to right. Strikes from the heel and toe must be accounted for when judging spin.

If a ball was to start spinning horizontally around its own axis there would be no forward movement, it would simply spin on the spot. If the ball was to start traveling upward and forward it would need back spin to lift through the air.

When a ball spins vertically around its own axis after being propelled forward by a club, the resulting shot will be straight. If the ball spun around its own axis towards the target it wouldnt lift, it would roll.

If a ball bounces left or right when it lands, the cause could be spin related or simply ground conditions. Spin should be judged by ball flight and not bounce. Side spin in its truest term does not really exist; it is the tilt of a balls spin around its own axis.