You're Correct!

The swing plane is the angle the club travels around your body. The swing plane is important during the golf swing as it affects both distance and direction. The swing plane is determined by the angle of the club shaft relative to the ball and the
ground. As the driver shaft is longer, you set up further away from the ball than you would with your irons. This makes the clubs shaft sit on a shallower angle, and therefore should be swung on a flatter swing plane than your irons.

The swing plane greatly influences club path, angle of attack and club head
speed.

Most great, consistent ball strikers can bring the club down to the ball on the same plane that they took it back on. The closer you can make your back swing plane to your down swing plane, the more consistent you will hit the golf ball.

A correct swing plane is made when you allow your hips and shoulders to rotate, whilst your upper body retains its inclination to the ground (no lifting or lowering).

If you drew an imaginary line from your ball and up through the shaft, this is the plane angle the club should be taken back and brought back down on. You can see this better if you take a video of your swing.

A good check point for you is to put another golf club on the floor along your toe line. Swing your driver half way back so that it is parallel to the ground. At this point your driver should be on the same line as the club on the floor. This is what the drivers position should look like at this point also on the down swing.

On the down swing ensure you drop your hands down so the golf club stays on plane and doesnt fire towards the ball which results in a plane angle that is too steep. Ensure your lower body is starting the down swing, and not just your arms.

Firstly, if you swing your driver back in a straight line you would not be able to swing very far back and therefore you wouldnt be able to generate enough club head speed to hit the ball a good distance. Secondly, the swing plane will be far too steep (vertical) for a driver, making you cut across the golf ball (out to in path).

By doing this you would be able to make a full back swing but the swing plane will be far too flat. This will encourage a swing path that is too far in to out which will enforce unwanted curvature, hooks and big pushes.

You may watch golfers such as Jim Furyk and see they have a loop at the top of the back swing. He has tremendous talent and has the ability to get the golf club in the right position at the right time. Therefore, you should focus on swinging back and through on the same plane.