I don't believe there is a separate name for either type of siege, they are both considered simply a siege. The whole point of a siege in general is to overtake the castle, and as you indicated, there are different means for going about doing this.
The considerations for choosing one over the other ultimately comes down to a basic function of time and resources. If you have plenty of resources (food, supplies, etc.) and are willing to wait them out, then a siege of starvation might be a viable option. This would ultimately resort in the lowest loss of life. The strategy is to simply outlast the defenders, who will have no way of replenishing their supplies because the attackers would have cut them off. The disadvantage of this is that it meant that you had to tie up a considerable number of men to keep the castle under siege. In the middle ages, a good portion of your army would have been peasants, and they were very likely to get bored and frustrated and longing for home, so this would be a risky tactic. In addition to that, it would become quite costly to keep paying for the supplies to provide for these troops, and before long the cost would become unreasonable.
The alternative, therefore, was to attack the castle by any means available to try to get it to fall. This would usually consist of a concentrated assault on one particular wall in an effort to either weaken the defenses there or overun them. The ultimate objective would be to breach the castle and either force the occupanst to surrender or simply kill them and be done with it. This option was much more common because it was more expedient.
Since each involves dramatically different conditions, its hard to say one would be easier to defend against than the other. The first option I listed would likely resort in a lower loss of life, so it might be preferred from a defensive position. As far as defending against an all out attack, it would really depend on what kind of castle you have and what resources you have within it. Some castles would be easy to defend because they were built to last, while others were mostly built for show, and therefore not easily defended.