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