Having operated out of ANC and LAX for many years now in widebody heavy aircraft, I have to say that I have not heard the expression "cleared high speed climb" at either location, but I am pretty sure that if they do say that they aren't meaning what you think they are.
Many of the heavier aircraft require climb speeds in excess of 250kts. Although aircraft are not required to request clearance for those higher speeds, many pilots do so as a common courtesy. I would guess that ATC is anticipating that and just saying that they know and approve of the higher speeds. Unfortunately, the meaning is different in other parts of the world, so I would not be surprised if some aircraft are climbing at much higher speeds.
The statement "normal speed" means just that, and does not restrict you to 250kts if you require a higher speed. The only thing that will do that is the statement "maintain 250 kts".