You say 'depending on the (virtual) die size.
I meant the effective die size 'seen' by the lens, the size of the virtual die image created by the dome or a pre-collimator. I didn't mean the virtual image created by the examined lens.
Your goal is a parallel beam. Thinking backwards, all the parallel rays go through the lens and (assuming a perfect lens) meet at the focal point, where the LED should be. Now if the lens has some distortion at some area (i.e. the edge), a parallel ray through that lens spot misses the focal point - but it might still hit the die, and that depends on the die size.
If it doesn't hit the die, that area of the lens is lost and doesn't contribute to throw.
If it still hits the die, everything is ok, since there is a part of the die sending rays through that lens spot which contribute to the parallel beam.
Thus with lens aberrations, a bigger die helps increasing throw.
I just wrote 'die' in the previous paragraphs, but it doesn't matter if that's a real, bare die or actually an 'apparent' virtual die created by a pre-collimator or the LED dome (which is a pre-collimator, too). That pre-collimator creates a magnified virtual image, i.e. a bigger virtual die, and that's why a pre-collimator can help increasing throw if the main lens has aberrations.
For a controllable focus and even spot, we want the same focal length I assume.
Yes, since that means all parallel rays behind the lens met at the same focal point, where the LED should be. With the Edmund lens, center rays intersect at some point (the focus), while outer rays intersect behind the focus.
But what does the magnification factor tells us?
Nothing relevant usually: The magnification in shots of that kind depends on the particular distance between lens and paper. A non-uniform magnification (like with the Edmund lens) however tells something about distortions and aberrations.
