TsumeGo Training can be divided into 4 different types that train different parts of reading:
This is basically the same as seeing the 'vital point', although more accurately it becomes seeing the 'vital sequence' as you advance. The idea is to know shapes so well that you immediately know the moves necessary to live/kill just by looking at the shape. The purpose it serves is to improve reading by being able to quickly discard/confirm sub problems in more complex problems. It is best trained by solving easier problems quickly.
This is the basic tsumego reading that was described in the quoted text above. Basically read all variations until the correct answer is found.
This goes one step further by not stopping once the correct answer has been found. Instead also study the other possible moves to better know why they don't work (or find other solutions if they exist). This allows for a more complete understanding of the problem.
After having done the above steps it can be nice to play it out on a real board. It allows the mind to see the variations on the real board instead of just in the mind. For problems that pushes the limit of ones reading ability, it also gives some more certainty that the answer is correct.