[YodasLuke]

When you see a curved line, does that mean you physically try to putt the ball in a curved line? I don't think so. Instead, you still make a STRAIGHT PUTT, but the slope is what causes the ball to curve.

For those curved liners out there, whether you realise it or not, you have automatically seen the straight line once you have visualised the curved path of the ball. The straight line is simply the starting direction of the curved path.
Vice versa for straight liners. You cannot know where to point your straight line unless you can visualise how the ball is going to curve into the hole. By seeing one you see the other.

I see both, and as a result will make friendly wagers with anyone on the putting green.

One of my best friends says he cannot see the curved path of the ball when he's putting. He understands the elementary fundamentals of slope and gravity, and he takes them into consideration in assessment of the putt. But, he cannot see the curved shape when putting. His lowest 4 day total in an event is 15 under par, so what does he know (straight liner)?

Anyone that can putt it into a hole smaller than the Grand Canyon understands that line is first and distance is second. By getting the initial direction first (straight line for Clubface and Plane Line tracing), then seeing the distance needed (Power Regulation on the curved line), putts can be made. This separation of duties is a must.

If you've done your homework (practice), then you should be able to trace a straight line with a Hinge Action. If you're playing, you should be able to visualize the ball going into the hole.

The practicer or technician, doesn't like the swirling wind that wreaks havoc on his trusted Venturi Effect, when hitting a shot into the air. The slopes on the green are out of his control, as would be the swirling wind. He doesn't like to see something knocking his ball off line.

The player or artist in putting sees the slopes on the green as if he could judge the swirling wind affecting a full shot. He permits gravity to have its way with his ball. Technicians don't like the forfeit of control.

I think that there is a blend of technical prowess and art that makes a great putter. IMHO, it's also why two of the greatest technicians in the game were driven to the brink of insanity when putting: Moe Norman and Ben Hogan.