Elastic bending means that the pipe is curved without exceeding its yield strength at any point. If you pick up a 12 meter length of pipe by hoisting it at the center, the ends droop down. If you lower it back to ground, once on the ground again it will resume its straight shape, without having to jump on it. It was curved within its elastic limits.
If you bend the pipe into a curve and it won't return to straightness after you've released it, you have yielded the pipe and have made a cold bend. Cold bends are usually not very sharp bends, most pipe will need minimum radaii of 40 x diameter.
Hot bends are made by heating the pipe to lower yield stresses, thus can be bent to tighter radaii.
Disadvantage of elastic bending is that if the pipe is bent elastically and held in that position forever (after construction has been completed), the elastic stresses, although less than yield stress) remain in the pipe forever. When you pressure the pipe, the pressure stresses will add to the elastic stresses, which could cause allowable stress to be exceeded when under pressure. That could require that your maximum allowable operating pressure be reduced, so the combined stresses remained below limits.
Cold and Hot bends have no remaining stress once the bending forces are released. Therefore hot and cold bends can be pressured without adding to any previous stress resulting from holding the elastic bend in place forever.