Two, and a leftover piece shorter than 2m. Due to the fact that saw blades have a non-zero width, and you want to err on the side of cutting them slightly too long. What do I win?
I'm from ACME corp, and I would like to introduce you to our - no, your! - new water-jet cutting machine!
It cuts anything, including that 6m board there with a cut so fine that you will be able to get 3 pieces of 2m out of a board of 6m, there is so little waste that within the acceptable margin of error for woodworking the cuts will be 0 width.
Sold?
If you require the pieces to be exactly 2m long without allowing for any margin of error, then there isn't a saw that will cut that accurately, so the answer is 0, not 2.
If you allow for the accuracy of the saw, then the answer is 3 since any saw can cut within its own accuracy. :-)