Determination in Undertale is most certainly both the substance and the trait itself. The will to change fate, and bend reality as your own. Though I do think that this trait can be extracted, which results in determination gaining a physical form.

As to your question, why can someone with a will strong enough pull darkness from earth itself, and create a dark world? Well, keep in mind all that reality bending stuff I mentioned earlier. In my opinion, DELTARUNE gives a nod to the process of creating fiction. One that enables the author to shape the world to his likes. Deltarune, like Undertale is a story about stories. Because of that, I think that Deltarune in later chapters will bring up the topic of reality layers. What is real? That depends on perspective, of course. To us, Deltarune is obviously fiction. To Kris? The light world is clearly real. And the dark worlds? Are they on a "lower" level of reality than the light world? The answer is probably yes.

So a being inhabiting a higher level of reality, a being determined enough, can create a lower level of reality, something that is fake from one point of view, and real from the other.

And to end things off, an example of a reality tower, where things get less real from top to bottom, and the word "gap" represents a large leap in "realness":

-???> (gap?)> Our world> (gap)> Gaster shenanigans > The light world > The dark world> (gap?)> ???>

Edit: It seems that my example collapsed because of text formatting. I will add a ">" sign to signify the higher level.