Path, plane and arc
The ball goal is to start the ball within 1 deg of your intended start line. The two impact features predominantly responsible are path and face. Path (yellow line in video) is the direction the whole putter is moving in. Face (black lines in video) is the angle the clubface is relative to the target.
Key concept - Where the face of the putter points at impact is five times more influential than the path on the ball’s start direction.
So face control is the primary skill.
Path plays its part. It can complement the face - e.g. if your tendency is to leave the face 1 deg open, this can be offset with a left path. However, if the difference between the two becomes too great, the glancing blow this causes creates excessive and unpredictable sidespin.
Because you are standing to the side of the ball, with the putter angled to the ground (e.g. 70 degrees not 90 degrees like a croquet mallet) it is easier and desirable to swing the putter on some form of arc.
The video below is to help you conceptualise how the plane of the putter creates an arc. The more upright the plane, the shallower/straighter the arc would be and visa versa.
It’s worth noting also that the arc is not just directional (curving away from and back towards the target line). As you can see there is also a gentle rise away and fall back to the ground, which is relevant to rise angle.
Having seen a putter move in plane, tracing a natural arc on the ground, it helps highlight the potential problem for a golfer whose concept was to take the putterhead absolutely straight back and through. This requires a more complex set of movements and is thus harder to repeat.
How much curve of arc depends on set up, putter lie angle, your mechanics and it’s quite individual. What’s key is to find what is most effective and repeatable for you. Knowing the goal is to bring the face back square at impact, it is also preferable to encourage a stroke that keeps the face square to the arc for most of the stroke. As per the picture below the putter face opens and closes to the target line in the stroke, but is still square to the arc the putter is travelling on. This means you won’t need to manipulate the face in the last few inches into impact.