You're Correct!
When you make a swing with any of your clubs, the way that the club exits the impact and follows through to the finish position is very important.
It not only ensures that the stroke you are trying to make has a good chance of being a successful one but it can also tell you a lot about what happened in the moments leading to impact with the ball.
For example, if you make a stroke and you pull your ball to the low side of the hole then you would see your putter face looking a lot more towards you, and if you trace your putter back to where impact would have been you are more likely to see a closed putter face.
One of the best ways to make sure that you keep your putter traveling through to a nice finish is to break the length of the stroke down into two thirds. The first third being your backstroke, then the last two thirds being the follow through after the point of impact. All the time you are thinking about moving the head of your putter twice as far after impact as it moved from address to the end of the backstroke.
A great way to practise this is to get either a yard stick or even an alignment stick. Place some tape in the centre, then another piece around six inches to the side of the centre and then one more piece around 14 - 18 inches the other side (if you have enough room). Practise making putting strokes over the stick, taking care to move your putter head back six inches and through 14 to 18 inches.
You need to practise this and count in your head nice and smoothly - 1, 2 and 3. Your backstroke will be 1, followed by impact at number 2 and then finally your putter stops moving at 3. Move the stick and just hit some putts, keep counting as you know if your count is good then your putter will follow through the right distance.
Sorry Try Again! - See Explanation Below
If your putter stops too soon after impact then you will see your putts stop short of the hole 9 times out of 10. This is because your putter will be slowing down in the forward stroke before impact. A good follow through will ensure that the tempo of your stroke remains constant to avoid leaving putts short.
Sorry Try Again! - See Explanation Below
As golf is a game played from a side on position, the arc of your putting stroke should see your putter face finish looking ever so slightly left of your aim line. If you try to keep your face looking at the hole then you will be likely to push the majority of your putts.
Sorry Try Again! - See Explanation Below
The follow through is just as important as any other aspect of the putting stroke because without a good controlled follow through, you will never have control of the speed or direction of your putts. This will see you making plenty of three or four putts on a regular basis.