One source indicates that it is the act of adding unfermented wort into a almost completed batch of beer. Is that correct?
That’s not Krausening but “Speisegabe”. Speise could be literally translated by “food” (for the yeast), Speisegabe by “feeding”.
Purpose: adjusting extract (gravity) for carbonation — primarily for wheat beer. In Germany priming with sugar is generally allowed only for top fermenting beers, but with the exception of Bavaria, Wuerttemberg and Baden, where any addition of sugar is forbidden.
The other indicates adding actively fermenting beer to a beer. Which is it?
The latter! Kräusen (Kra:usen, Kraeusen) has two meenings:
A) Rocky head during fermentation.
B) Fermenting beer at high krausen — usually 35-45% apparent attenuation.
So in any case it is used in conjunction with fermentation. I’ve seen German texts that extend the term Speise to include Krausen (eg. something like …first runnings, knock out wort or _krausen_ can be used as Speise… ) — seems logical, krausening is also some kind of feeding. But I never came across a text that uses Krausen for unfermented wort.
What is the real purpose?
Fermenting beer is used in several ways like eg. pitching with krausen etc. but I think the term krausening refers to the German term “aufkra:usen” witch has the same purpose as Speisegabe (carbonation). It’s sometimes preferred because the fresh yeast gives a better attenuation. — Note that in Germany generally an apparent attenuation over 80% is striven for. Despite popular believe body and full mouth feel (fullness?) are primarily determined by the colloids (proteins) and not unfermented or unfermentable sugars!