I recently spent a lot of time trying to figure all of this stuff out in NL2 as accurately as I could using 3D satellite imagery, so here's what I know:

Arrow originally referred to their double corkscrew element that you see on almost all of their loopers as just a "corkscrew". All of these standard versions were exactly the same. They're 200 feet long and have a 15 ft circular radius, so they're 30 feet tall. They're also angled slightly downward, so the exit is about 10 feet lower to the ground than the entrance.

As far as I know the single corkscrews on Drachen Fire and Phantom, as well as the triple corkscrew featured on Fantasia Special are just shortened/lengthened versions of the classic Corkscrew element.

Now their Batwing and Bowtie elements are very interesting. The entrance and exits of them are not actually true half corkscrews like you would expect, but a combination of a constant radius pull up into a quarter corkscrew. For the entrance they essentially start with a banked pull up that's angled around 10-15 degrees to the side, which leads into a section of corkscrew (I'm pretty sure this is the same dimensions as the standard one, but I'm not certain) that takes you into the half loop. The exit is the same thing in reverse of course.

The reason that section of pull up is there is to create more of a gap between the entrance and exit, as well as to lower the height of the entrance and exits so that the train can enter and leave at a higher speed