Keywords are put on a power, based upon the damages and effects that can be caused by the power, and the general nature of the power. The damage types within the power define the keywords, not the other way around.
Blazing Starfall does not on its own cause fire damage. However, a Sorcerer with the Cosmic Magic class feature can use the additional text under that heading in the power. This creates a zone of burst 1 until the end of the Sorcerer's next turn, from which an enemy cannot leave without taking fire damage.
This is why the Fire keyword is part of Blazing Starfall. The Troll will not take actual Fire damage unless the Sorcerer has the applicable class feature, and the Troll moves out of the zone prior to the start of the Sorcerer's next turn.
It is noteworthy though, that this power may perhaps be affected by other feats and effects that target the Fire keyword itself rather than Fire damage. This is under some debate however. (See comments on the other Answer in this thread.)
Having answered the general question, here's how it applies to your specific scenario:
Since your Sorcerer was not a Cosmic Magic Sorcerer, they could not have created the zone effect available through Blazing Starfall. Therefore, there would not be any opportunity for the Troll to trigger any Fire damage. So, there would be no means for your Sorcerer to disable the Troll's regenerative abilities through Blazing Starfall.